text: https://www.enisa.europa.eu/publications/post-quantum-cryptography-current-state-and-quantum-mitigation, (accessed October 17, 2022). e.g. NIST’s https://csrc.nist.gov/Projects/post-quantum-cryptography, (accessed October 17, 2022). code-based, isogeny-based, hash-based, lattice-based and multivariate-based https://csrc.nist.gov/Projects/post-quantum-cryptography/round-3-submissions, (accessed October 17, 2022). viz. hybrid implementations that use a combination of pre-quantum and post-quantum schemes, and the mixing of pre-shared keys into all keys established via public-key cryptography. While this report was being typeset and proofread NIST announced (July 2022) it had identified the four candidate algorithms for standardisation https://csrc.nist.gov/News/2022/ pqc-candidates-to-be-standardized-and-round-4 and https://csrc.nist.gov/publications/detail/nistir/ 8413/final, (accessed October 17, 2022). See for example https://www.scientificamerican.com/article/ will-quantum-computing-ever-live-up-to-its-hype/ and https://spectrum.ieee.org/ the-case-against-quantum-computing, (accessed October 17, 2022). As shown by the two recent cryptanalysis attacks against the Supersingular Isogeny Diffie-Hellman (SIDH) protocol – one by Wouter Castryck and Thomas Decru of KU Leuven and the other by Luciano Maino and Chloe Martindale of the University of Bristol. SIDH is at the core of the Post-Quantum key encapsulation mechanism SIKE (Supersingular Isogeny Key Encapsulation), which was selected to continue to round four of the NIST Post-Quantum Project for consideration of standardisation. https://eprint.iacr.org/2022/975, https://eprint.iacr.org/2022/1026 (accessed October 17, 2022). 9 https://bench.cr.yp.to/, (accessed October 17, 2022).

text: , https://www.enisa.europa.eu/publications/post-quantum-cryptography-current-state-and-quantum-mitigation (accessed October 17, 2022). https://csrc.nist.gov/News/2022/pqc-candidates-to-be-standardized-and-round-4 and https://csrc. nist.gov/publications/detail/nistir/8413/final, (accessed October 17, 2022).

text: An exception are stateful hash-based signatures such as LMS and XMSS. These already standardised schemes take a special role as they do not match the standard signature API because they require the secret key to be changed after every signature. However, their performance is similar to that of structured lattice-based schemes.

text: mceliece69 20

text: T: • 19

text: implementation. em1380

text: this rodokem976(aes,shake) • ntsk r f kem34sl mceliece66 T: T: kem54sl 18 fo rodokem640(aes,shake) leda 6(,f) • 2

text: f leda T: T: T: goal dakem52sl dakem3464 bears936r2cpax a le le ars1248r2cpax 3 qc2562 T: dakem33sl T: skem1264 C: h le er9 3be nt as T: T: olloii192r C: edakem1464 mceliece46089 T: akem13sl lock l akem5464 led T: T: T: time odokem976 T: edakemcpa33l 17 fr rodokem640f led rs936r2cpa, 2 T: T: C: rodokem1344(aes,shake) T: T: 3bea f ars1248r2cpa, C: 4(,f) T: 1024 3be constant akem53sl ikel3 C:

text: led b oi256 list T: C: edakemcpa14l 16 c256 T: kyber(,90s) roll C: mceliece34886 2 hq 5kem0d dakem1364 C: not T: r5n1 le loii128 T: T: akem5364 kemcpa32 kem14sl T: rol a,3bears936r2ccax, es C: sntrup1013 T: led leda leda T: do 13kem0d T: C: T:

text: r5n T: el1 C: 2cca,3bears1248r2ccax, bik C: ledakemcpa13 3bears936r2cc 68 15 T: akemcpa34 C: 2 T: database ledakemcpa53 led T: C: C: 3bears1248r 5nd5kem0dr T: T: C: kyber(,90s)7 • edakemcpa54 cker3 T:

text: qc2563h l lo T: • C: T:

text: T: sntrup857 trup4591761 14

text: •• sn 2 SUPERCOP • • dakem5264

text: qc2561h le • T:

text: T: T: the qc1922h • ears624r2cpax T: • • sntrup953 21 3b qcrmrs256 •• 11kem0d kem3364 C: sntrup761

text: Project that • h • •

text: •• T: r5n leda • ntruhrss701 13 192 • • C: T: • 2 hqc • T: • sntrup1277

text: dakem3264 ntruhps40968 rs624r2cpa, means T: • le • loi192

text: rdization ” • T: • rol 3bea • • T: •• C: C:

text: “ • • •• trukem743 T: sntrup653

text: Standa • T: 5nd3kem0dr 12 921 edakem32sl • • 2

text: l C: hqc1 T: • T:

text: rimitive. T: 56 crmrs192 • • •

text: p hq • • ,3bears624r2ccax, rqc2 T: • • • ntruhps2048677 dakemcpa52 •

text: this T: le •

text: r 1281 C: • 11

text: Cryptography fo hqc • • • 2 T: • •

text: T: • ker8 • • 12 3bears624r2cca

text: oal loc • • • • ke3

text: g 128 • • la a hqc qc192r qc128 T: •• C:

text: r ••• •• •

text: ). as T: T: T: • • T: oii256 ••• • •• kyber(,90s)5 3kem5d

text: ). • ost-Quantum ty roll • • ••• • 10 P • •• • • r5nd 2

text: kem12sl edakem1264l T: • • • C:

text: curi • T:

text: se crmrs128 leda T: • • • hq T: ker7 ••

text: NIST T: loc •• r • • fo T: • •

text: ila5 • • ocker1l 9 9

text: hope1024cca h T: 2 new C:

text: IND-CCA2 T: •• •

text: T: •

text: excerpt list ntrulpr1277 •

text: size, not ac128l 1 crypto_kem_PUBLICKEYBYTES ntrulpr1013 lac

text: ( es ker6 c256 T: T: • 8

text: crypto_kem_CIPHERTEXTBYTES la ntrulpr953 lake ep751 2

text: ( do loc T: C: key T: • T: sik

text: T: er5 d5kem5d • ciphertext resaber2 er4 r5n lock fi ocker2 public T: T: l lock C:

text: size, a database T: T: T: r ciphertext kep751comp 7

text: fo a si ikep610s 2 key r dakemcpa12 er2 nd1kem0d d1kem4longkey T: , le r5n T: es) fo sab ightsaber2 r5 2 nd1kem5d C: ntrulpr857 lpr4591761 T: l C: C: yt T: ntrulpr761 hope512cca T: lloi128 T: lake T: r5 0kem2iot (b ntru ro C: C: p503 ytes) new C: ukem443 T: r5nd kep434 (b SUPERCOP T: T: T: sike C: ikep610comp ikep503comp si T: ntr T: s s kep434comp T: ntrulpr653 T: T: T: T: si 6

text: crypto_kem the 2 Space T: , s: Space that xia

text: hiphop axis: means 5 rizontal tical ” 2

text: er C: 15 14 13 12 11 10 9 8 7 6

text: amd64, Ho V “ 2 2 2 2 2 2 2 2 2 2

text: https://bench.cr.yp.to p610

text: akemcpa52 ep751 sike 27 dakem53sl T: 2 akem52sl le led sik led edakem5364l C: T: qc256r T: T: T: T: T: dakem5464 akem5264 le el3 led T: bik dakemcpa54 • •

text: T: C: le kep503 T: C: si 26 kem3464 T: T: 2

text: akem54sl led leda ep751comp odokem1344shake T: T: • sik fr cker9 • T:

text: lo ikep434 T: T: s dokem1344aes ker8 T:

text: fro loc ep610comp ikep503comps 2

text: T: okem976shake kemcpa53 T: sik er6 T:

text: implementation. frod leda T:

text: T: C: lock T: • this • T: cker3 • kep434comp • • er5 lo si r kem3264 T: fo • er7 lock T: 24 • leda akem34sl T: e3 2 T: led lock dakemcpa33 ocker4l • goal • dakem33sl T: T: le • lak • edakem32sl le C: T: C: a l T: T: T:

text: T: ocker2l

text: as • • T: •

text: • cker1 ake2l

text: • lo 23

text: time rqc • C: 2

text: T: • • T: T: • • kem1464

text: rodokem640shake • • • • • leda f • kem13sl kem14sl • ake1l loi128

text: constant T: dakem12sl qc128 T: C: rol dokem976 le r leda leda akemcpa34 C: 562 T: T: T: T: T: list fro dakem3364 • olloii256 • led • dakem1364 T: 22 T: le r • C: le 2 hqc2 T: • T: not •• T: T: T:

text: dakem1264 akemcpa13

text: es le •

text: • T: • • lloi256 led

text: do • • • ro C: •

text: • oi192 C: T: c2561 • • T: tskem13136n rodokem976aesf • • hq • roll 21 T: • • • C: T: 2

text: • T: • • • •

text: • • • T: skem1380 19f

text: database • nt

text: akemcpa14 tskem1264 led T: n • el1 • C: T: • bik 128f • odokem640aes C: • T:

text: fr 2128f 2128 mceliece69601 • kemcpa32 T: 20 dokem640 T: • 28 2 2563 rmrs256 fro leda 1281 • • SUPERCOP hqc C: hqc loii128 mceliece6688 hqc T: T: T: T: T: rol • mceliece819 mceliece819 the qc256h T: • crmrs192 • T: • loii192 Project c1921 hq cca ac256 rol 5kem0d rukem743 mceliece66881 0119 96 that hq T: ccax l ac192l nt 5kem5d c192 mrs128 T: T: r5nd T: 19 hq T: akemcpa12 T: C: r5nd nd3kem0d 2

text: T: hqcr led T: C: r5 means T: C: T: C: mceliece696 mceliece4608 rdization qc128 3bears1248r2 T: T: • ” 922 h • 3bears1248r2 •

text: T: T: a ukem443 • “ hqc1 7 • ntr cax ••• 5nd3kem5dr T: hope1024cca •• C:

text: Standa T: •• 18

text: • new T: 2 15kem0d T: • • f

text: r5n 13kem0d ntrulpr127 • 3bears936r2c rimitive. C: r5n iresaber2 11kem0d 3bears936r2cc • • • • 2cca

text: p C: f r5n T: T: • ). ntrulpr953 T: C: •

text: this T:

text: Cryptography r • • • mceliece460896 3bears624r 17 fo ntrulpr1013 •• 2

text: • • • • • oal • • g • •

text: a as • • • • • crypto_kem_dec ). • ulpr4591761

text: ost-Quantum ( ty ntrulpr857 • • ntr 2ccax 16 P • •• mceliece3488 2 • T: mceliece34886

text: curi • • • •

text: se sntrup1277 ac128 3bears624r

text: NIST l • r ciphertext T: • • •

text: fo • nd0kem2iot a ber2 1 • r5 hope512cca sa C: 15 new T: • T: 2 IND-CCA2 T: sntrup1013 • •sntrup653

text: excerpt given • list

text: key la5 ntruhps409682 • size, sntrup953 hi 1kem4longkey not C: • nd1kem5d es T: sntrup857 • r5 r5nd 14

text: session crypto_kem_CIPHERTEXTBYTES 7 C: C: 2

text: ( do ghtsaber2 T: T: a ntruhrss701 li

text: T: ntrup4591761

text: ciphertext kyber1024 s

text: ntrulpr761 T: ntruhps204867 5nd1kem0d

text: time, database rs1248r2cpa r generate ciphertext ntrulpr653 509 C:

text: a 3bea T: 13 dec to r C: ars1248r2cpax kyber512 2

text: , fo 3be ars936r2cpa ars936r2cpax

text: C: 3be 3be bears624r2cpax3 kyber90s1024 sntrup761 ntruhps2048 C: C: kyber768 C: bears624r2cpa3 (cycles) ytes) SUPERCOP kyber90s768 C: kyber90s512

text: (b the 12 crypto_kem Time 2 , s: Space that xia

text: hiphop axis: means tical ” 11 rizontal 9 8 7 6 2

text: er C: 15 14 13 12 11 10

text: amd64, Ho V “ 2 2 2 2 2 2 2 2 2 2

text: https://bench.cr.yp.to 34 2

text: mceliece66881 mceliece8192 128f mceliece696 30

text: 119f edakem5364l implementation. mceliece6688 T: edakem52sll ikep751comps 29 T: dakem1264 ep751 T: 2 • edakem3264 le ep610comp this • 6 96f dokem976shake edakem3364 l T: ocker8l sik

text: r fro l T: T: sik • mceliece6960 odokem1344shake T: T: T: fo • T: fr ocker7l 28 • •• 64 T: loii192 T: 2

text: kem5464 rol goal mceliece4608 cker5 mceliece46089 leda kem53sl T: lo ep503comp a T: T: er4 sik • • leda er9 T: 27 as 8f mceliece3488 rodokem1344aesf akem54sl T: lock kep503 2 • T: lock T: si

text: led akem34sl akemcpa52 T: T: time T: kem5264 dakem12sl oii128 led led le ker6 ep610 kep434comp T: leda C: T: roll • si 26 okem976aes T: loc sik T: 2 • T: T: T: T: mceliece819212 • frod edakem32sll skem13136 T: • ocker1 •

text: constant nt T: ocker2 l 25 T: • • dakem33sl l T: 2 • le T: list tskem1380n kem1264 rodokem976 T: • T: f lloi256 ep434 ake2 nts T: • • oii256 ro sik • l not T: • C: T: C: 24 es kem14sl rodokem640shake • • roll • ker3 T: T: 2 f kem3464 • • T: • • loc • do leda T: • ake3l • ke1

text: 8864f T: leda T: la rodokem640aes T: •• • • sntrup1277 C: C: dokem640 f • • T: 23 T: • • kem743 • T: 2 fro • • • loi192 akemcpa12 edakemcpa54l T: • • • ntru rol led database mceliece34 C: • • • T: C: C: olloi128 • • sntrup857 r 22 T: •• • • T: T: C: • • sntrup953 trup4591761 2 • kem13sl • sn T:

text: leda • • kem1364 • • 21 T: sntrup1013 • 2 leda • akemcpa14 • • SUPERCOP dakemcpa53 T: • c192 led •

text: le rq ukem443 dakem1464 C: 5n15kem0dr • C: • nd3kem0d the le T: T: •• • ntr r5 20 T: T: C: • sntrup761 77 T: C: 2 akemcpa32 T: •• qc256 • • 821 09 kemcpa33 •• • r • T:

text: Project that led kel3 • T: • • •

text: C: leda •

text: T: bi 19 C: C: • • 5nd5kem5dr 2 T: T: •• 281 • ntruhps4096 C: ntruhps20486 means akemcpa34 2563 • • kel1 sntrup653 3kem5d ntruhps20485

text: hqc1 ntruhrss701 T: bi ” led 3kem0d hqc qc128 T: c128 C: r5nd rdization C: T: crmrs256 h rq • T: d1kem4longkey 18 T: hq T: T: • C: 2 “ T: r5n1 256 T: • • • T: r5n C: 2562 n11kem0d • • • C:

text: T: hqc hqc r5 T:

text: Standa T: • •

text: T: C: nd5kem0d •• 17 T: c1921 r5 •• 2 hqc2561 hq C: • • • •

text: rimitive.p T: crmrs128 •

text: T: dakemcpa13 hq T: ••• c1922 qc192 le la5 T: •

text: h C: • • •• 16 this hq T: hi • • 2

text: r T: mrs192 T: C: • • •

text: T: • • Cryptography fo hqcr • • ••• T: • • • • •

text: oal ears1248r2cpax • ac192l d0kem2iot 15

text: g 3b T: r2 2 a C: • sabe r5n nd1kem5d

text: uantum • C:

text: ). as T: T: r5

text: -Q C: ). ty • T: 14

text: ntrulpr1277 nd1kem0d 2 ost 256 r5 P curi r2cca lac ax C: cax ope1024cca kyber1024 T: se T: a bears1248r2cpa newh 13 NIST 3 T: esaber2 2 r C: 3bears1248 fo fir x 3bears1248r2c T: ntrulpr761 3bears624r2cc ears936r2cpax ears624r2cpa ac128l 12 IND-CCA2 3b 4 3bears936r2cc kyber90s768 2

text: crypto_kem_keypair C: 3b T:

text: ( C: cca ightsaber2l

text: excerpt list T:

text: key ntrulpr1013 ntrulpr953 3bears936r2cca 11 not kyber90s102 bears624r2cpax3 2 size, ewhope512cca es 3bears624r2 C: n

text: public crypto_kem_PUBLICKEYBYTES kyber90s512 ntrulpr653 T:

text: key ( do rs936r2cpa kyber768 10

text: 3bea ntrulpr857 kyber512

text: key C: time,

text: rulpr4591761 nt 9

text: database T: 2 generate public to a keypair r , fo 8 2

text: (cycles) ytes) SUPERCOP (b 7 2 crypto_kem the Time , s: Space that xi 6 a 2 hiphop axis: means 5 rizontal tical ” 2

text: er C: 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6

text: amd64, Ho V “ 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

text: https://bench.cr.yp.to 26 2

text: dgemss256 luegemss256 s256 gems re b T: 24 T: T: gems 2 dgemss256v2 T: re T: emss192g bow5c egemss256v2 T: 23 dgemss192v2 blu luegemss192 2 rain re T: b v8117404pc T: T: dgemss192 T: re luo T:

text: T: 22

text: ov890351pc mss192v2 ••

text: lu ge ••••

text: T: T:

text: ainbow4a • 20

text: rainbow3c T: mss128 T: bow6a ge inbow6b • T:

text: ra rain edgemss128r T: T: • • T: uegemss192v2 19 bl 2

text: inbow3b • ••

text: T: 18

text: 8086399 863256pc •

text: tesla3pq luov luov 17 T: T: T: 2 uegemss128 • bow1c

text: bl rain T: ov890351 T: 16 lu 2 T: ss128v2 inbow1b gem tesla1pq • ra T: • • T: T: emss128v2

text: le) 15 luov6468330 uov8117404 inbow1a redg 2 T: l ra e) la3f 4849242 T: T: T: uegemss128v2 bl qtes • luov T:

text: T: T: 14

text: 6,shake256)(robust,simp sla3s 13 hake256)(robust,simple) ,aes) 2 • qte

text: Project ,shake256)(robust,simpl T: uov863256 lcon1024tree l dilithium3( fa T: 12 ic2l5fs T: 2 rdization picn ncss256(haraka,sha256,s •• phincsf256(haraka,sha25 T: •

text: s sphi 11 T: T: incsf128(haraka,sha256,s • 2 hincss192(haraka,sha256 ,aes) •• Standa sp sph con512dyn T: T: ) • • fal

text: ) T: 10 ss48 dilithium5( 2

text: ). mqd ••

text: dss64 T: aes) mq

text: implementation. T: 9 Cryptography cnic3l5 2

text: this T:

text: r shake256)(robust,simple dilithium2(, esla1 fo icnicl5fsp nicl5full qt T: shake256)(robust,simple T: 8 ). pic lcon1024dyn on512tree 2 T: fa goal T: falc

text: crypto_sign_BYTES a icnic3l1p T:

text: ost-Quantum picnicl5ur T: P as T: nic2l1fs 7 pic 2

text: most T:

text: NIST time • incss128(haraka,sha256,

text: r (at sph

text: fo hincsf192(haraka,sha256, T: 6

text: • • • •• • sp • • 2

text: rpt • • • • •• • constant exce message • • 5 list • • • • • 2

text: size, long a not crypto_sign_PUBLICKEYBYTES( g es cnicl3ur nicl1ur 4

text: n pi pic 2

text: do T:

text: key icnicl3fsp T:

text: signi T: signature r icl3full

text: fo nic3l3 3 public picn pic nicl1fs 2 size, a database T: T: pic nicl1full icnic2l3fsp

text: r ytes) T: pic T:

text: key fo (b T: , 2 2 es) yt (b SUPERCOP 1 overhead 2 crypto_sign the Space , s: Space that 0 xia 2

text: hiphop axis: means 1 tical ” − rizontal 9 8 7 6 5 4 2

text: er T: 19 18 17 16 15 14 13 12 11 10

text: amd64, Ho V “ 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

text: https://bench.cr.yp.to 34 2

text: hincsf192shake256robust 32 sp cnic2l3fs 2

text: icnic2l5fs T: pi p T: T:

text: nic3l5 pic ov8086399 30 T: lu 2 T:

text: hincsf128shake256robust sp nic2l1fs 29 T: 2 pic T: hincsf256shake256simple ncss256sha256robust sp incsf192shake256simple incss192shake256robust v6468330 T: • sphi incss256shake256robust sph luo 28 incsf256sha256robust sph sph T: T: 2 T: T: sph T: hincss256shake256simple T: hincsf256shake256robust sp icnicl5urp sp ncsf192sha256robust • T: ncsf128sha256robust T: T: • 27 sphi sphi phincss192sha256robust 2

text: icnicl5fs T: T: s p phincsf192sha256simples cnic3l3 T: T: phincsf256sha256simples T: pi T: T: hincsf128shake256simple icnic3l1p • incss128shake256robust ov4849242 8117404 890351 26 • sp lu 2

text: T: sph T: luov luov T: T: T: T:

text: nicl3ur •

text: pic • • T: • hincsf128sha256simple

text: • sp • 24

text: • • •• uegemss256v2 2 • T: emss256v2

text: • • • redg T: emss256v2g • • • cnicl1fs • T:

text: edgemss256r T: 23 pi • • • ov863256 nbow6b mss256 2

text: T: lu rai T: ge • •• T: T: T: cnicl5full • • •

text: pi • edgemss192v2 22

text: T: • • • r 2

text: icl3fs • • T: • egemss256 dgemss192 picn • • re T: • blu T:

text: • • • T: 21 • 2

text: Project e ainbow5cr • luegemss192

text: icnicl1ur T: inbow6a b nicl3full p •• incss128sha256robust ra T: ss64 pic T: • hincss192shake256simple sph T: ••••••

text: mqd T: sp T: 20 rdization T: phincss256sha256simple • T: • la3p tesla1pq 2 s T: v8117404pc • T: qtes • • • T: luo •••

text: • T: ••• Standa bow3b 19

text: qdss48 incss128sha256simple phincss128shake256simpl con1024dyn • 2 m ncss192sha256simple s mss128 T: sph T: fal rain • s192 ge phincsf256harakasimple phincsf128harakarobusts sphi T: T: • T: • s T: T: gems T: ). T: • uegemss192v2 • 18

text: •• bl T: • • 2

text: ). icl1full T: • • ••

text: picn • • nbow3c

text: Cryptography • rai inbow4a

text: implementation. hincsf256harakarobust T: • ra sp T: 17 T: hincss128harakarobust • • T: 2 hincss256harakasimple sp • • 890351pc mss192v2 this sp T: ge r incss256harakarobust sla3s • T: fo sph T: • luov qte • T: T: T: ov863256pc • 16 crypto_sign_open( phincsf192harakarobust 2

text: goal s lu

text: ost-Quantum T: esla1 T: gemss128

text: P crypto_sign_BYTES a dilithium5 qt inbow1b •

text: hincss192harakasimple ra red egemss128 ss128v2 as phincsf192harakasimples T: T: T: blu 15 sp gem 2 NIST message most T: T: tesla3fq gemss128v2 T: T: r phincsf128harakasimples T: red time T: aes T: fo (at dilithium3 es ainbow1cr egemss128v2

text: con1024tree T: blu 14 signed ncss192harakarobust fal T: 2 a ncss128harakasimple dilithium5 T: excerpt constant sphi dilithium2 con512tree message T: sphi dilithium3a T: fal 13 given list T: lcon512dyn nbow1a 2

text: size, fa rai

text: long dilithium2aes T: T:

text: a not gn es 12 message do 2 signature a signi r fo 11 time, 2 database generate open ytes) to (b

text: (cycles) SUPERCOP 9 overhead 2 crypto_sign the Time , s: Space that 8 xia 2

text: hiphop axis: means ” 7 rizontal tical 2

text: er T: 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4

text: cnic2l3fs pi

text: T: 34

text: sphi hincsf192shake256simple 33 T: sp icnic2l5fs 2

text: T: p ncsf128shake256robust T: hincsf128sha256robust ic2l1fs sphi sp 32 T: T: 2 tpicn T: icnic3l5p icnic3l3p 31 T: T: 2

text: incsf256shake256simple sph cnicl5fs phincsf256sha256robusts T: pi T: 30 T: 2

text: phincsf192shake256robuss 29 t T: 2

text: ov8086399 lu nicl3ur t T: phincss256sha256robust pic • hincsf192sha256robust 28 s nicl5ur sp 2 T: T: T: pic ov6468330 T: • 27 • lu ncss192shake256robust 2

text: T: phincss256shake256robus sphi icnicl3fsp phincsf192sha256simples s T:

text: • T: • 26

text: T: incsf256sha256simple T: phincss128shake256robus 2 ov8117404 hincsf128shake256simple s lu ov4849242 sph • sp T:

text: T: lu T: T:

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text: T: • •

text: inbow6a • 21 ra gemss192v2 qtes • • • • T: T: 2 red ncss256shake256simple T: uov863256l • •

text: •••••• sphi •

text: T: T: 20 Project nbow3c nicl1ur • • 2 • • • mss128 cnicl3full pic • rai ge dilithium5 pi phincss192shake256simpl T: ••••• T: sla3f • incss256sha256simple s ncsf128sha256simple • 19

text: nbow3b T: T: T: 2 • qte sph • rai T: dss48 T: sphi rdization • • T: mq T: phincsf256harakarobusts • T: T: phincss192sha256simple 18 ••• •• s icnicl1fs 2

text: ••• • T: p

text: ss192v2 • • • T: Standa ). ov8117404pc • phincsf192harakarobust ncsf256harakasimple phincss128sha256robust lu gem • • s s 17 dgemss192 egemss192 T: nbow4a tesla3s • T: sphi T: 2 T: re rai ainbow1b q • T: blu r T: • T: e T: T: T: • • ss128v2 dilithium5aes • 16

text: implementation. ss192 gem 2

text: gemss192v2 dilithium3 incss128harakarobust incsf128harakarobust gem dgemss128v2 • T:

text: Cryptography this T: blue re sph sph

text: r T: T: ncss256harakarobust T: T: 15 egemss128v2 2 fo sphi

text: crypto_sign_open blu falcon1024dyn es T: cnicl1full ( T: T: lcon1024tree tesla1 14 goal fa q pi phincss128shake256simpls ). T: T: T: 2 a luegemss128b T:

text: as edgemss128r T: uov863256pc dilithium2 ncsf128harakasimple ost-Quantum T: l inbow1c inbow1a dilithium3a 13 P message luov890351pc T: ra ra ncss128sha256simple sphi 2 T: T: T: dilithium2aes incss256harakasimple T: time sph sphi on512dyn ncsf192harakasimple T: T: phincss192harakarobust 12 NIST lcon512tree falc s 2 r signed fa T: sphi T: fo a T: T: phincss192harakasimples

text: constant T: 11 2 given list excerpt 10 not 2 size, es message crypto_sign_PUBLICKEYBYTES( do incss128harakasimple 9 key a sph 2 key T:

text: time, 2 database generate public open to a 7 , r 2 fo

text: (cycles) ytes) 2 SUPERCOP (b the 5 crypto_sign Time 2 , s: Space that xia 4 2 hiphop axis: means ” 3 rizontal tical 2

text: er T: 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4

text: amd64, Ho V “ 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

text: (accessed October 17, 2022). https://www.openssh.com/txt/release-9.0, Feature activation is the remote activation of features that are already implemented in the software and hardware of the car. For example, an additional – pre-installed but deactivated – infotainment package. A short activation code can be protected with a signature to prevent unauthorised activation of the feature.

text: CISCO has issued guidance on how to configure MACsec for post-quantum for their routers. See also [45] for the corresponding scientific paper.

text: https://www.cisco.com/c/dam/en_us/about/doing_business/trust-center/docs/ configuring-post-quantum-macsec-in-cisco-switches.pdf, (accessed October 17, 2022).

text: Daniel J. Bernstein, "Elliptic vs. hyperelliptic, part 1", pages 4-6, https://cr.yp.to/talks.html#2006.09.20, (accessed October 17, 2022).

text: This is not to say that double encryption/signature is a catch-all solution to badly written or malicious software. If, for example, due to a supply-chain attack, the new system uses a compromised software library with malicious code enabling remote code execution security overall will be compromised, despite the soundness of the cryptography in use.

text: It should be noted that we commonly aim for security against computationally bounded adversaries. That is, we require that it is hard to solve a problem or break a cryptographic system in reasonable time. In this setting, eventually the security of some basic building block has to be established by cryptanalysis. Security proofs can help by focusing the cryptanalytic efforts on a few selected problems. Videos and slides are available at https://quiques.huelsing.net/, (accessed October 17, 2022).

text: https://www.enisa.europa.eu/publications/post-quantum-cryptography-current-state-and-quantum-mitigation, (accessed October 17, 2022).

text: (accessed October 17, 2022). https://www.etsi.org/technologies/quantum-safe-cryptography, https://www.etsi.org/newsroom/press-releases/1805-2020-08-etsi-releases-migration-strategies-and-recommendations-for-quantum-safe-schemes, (accessed October 17, 2022). https://datatracker.ietf.org/doc/html/draft-ietf-ipsecme-qr-ikev2, (accessed October 17, 2022). 4 (accessed October 17, 2022). https://datatracker.ietf.org/doc/draft-uni-qsckeys/,

text: through standardization, such as the ongoing NIST PQC process . As an indication, from the 82 initial proposals in 2017, only four candidate algorithms have been selected in July 3 2022 for standardization and four additional algorithms will continue into the fourth round. So any investment in integrating the withdrawn or rejected candidates would have had diminishing returns. Any changes will need to be tested to ensure they do not unknowingly introduce undesirable changes; for example security weaknesses, both at the design level and for each implementation.

text: e.g. NIST’s https://csrc.nist.gov/Projects/post-quantum-cryptography, (accessed October 17, 2022). One of fourth round selections, SIKE, was cryptanalysed later that month https://eprint.iacr.org/2022/ 975. A second attack was published soon after https://eprint.iacr.org/2022/1026, (accessed October 17, 2022).

identifier: Post-Quantum Cryptography - Integration study

title: Post-Quantum Cryptography - Integration study

text: CRYPTOGRAPHY

text: Integration study OCTOBER 2022 POST QUANTUM CRYPTOGRAPHY Integration study October 2022

text: ABOUT ENISA

text: CONTACT

text: To contact the authors please use evangelos.rekleitis@enisa.europa.eu For media enquiries about this paper, please use press@enisa.europa.eu

text: AUTHORS

text: Daniel J. Bernstein, Ruhr University Bochum Andreas Hülsing, Technische Universiteit Eindhoven Tanja Lange, Technische Universiteit Eindhoven

text: FOR ENISA

text: Evangelos Rekleitis, ENISA

text: ACKNOWLEDGMENTS

text: Nikolaos Tantouris, ENISA Radu Arcus, ENISA

text: LEGAL NOTICE

text: This publication represents the views and interpretations of ENISA, unless stated otherwise. It does not endorse a regulatory obligation of ENISA or of ENISA bodies pursuant to

text: Regulation (EU) No 2019/881

text: . ENISA has the right to alter, update or remove the publication or any of its contents. It is intended for information purposes only and it must be accessible free of charge. All references to it or its use as a whole or partially must refer to ENISA as its source. Third-party sources are quoted as appropriate. ENISA is not responsible or liable for the content of the external sources including external websites referenced in this publication. Neither ENISA nor any person acting on its behalf is responsible for the use that might be made of the information contained in this publication. ENISA maintains its intellectual property rights in relation to this publication.

text: COPYRIGHT NOTICE

text: © European Union Agency for Cybersecurity (ENISA), 2022 Reproduction is authorised provided the source is acknowledged. This publication is licenced under CC-BY 4.0 “Unless otherwise noted, the reuse of this document is authorised under the Creative Commons Attribution 4.0 International (CC BY 4.0) licence (https://creativecommons.org/licenses/by/4.0/). This means that reuse is allowed, provided that appropriate credit is given and any changes are indicated”. Cover image © shutterstock.com For any use or reproduction of photos or other material that is not under ENISA copyright, permission must be sought directly from the copyright holders.

text: ISBN: 978-92-9204-590-6. DOI: 10.2824/151162, Catalogue Number: TP-03-22-080-EN-N

text: i POST QUANTUM CRYPTOGRAPHY Integration study October 2022

text: CONTENTS

text: 1. INTRODUCTION 2 2. INTEGRATING POST-QUANTUM SYSTEMS INTO EXISTING PROTOCOLS 4

text: 2.1 SIZE AND SPEED OF POST-QUANTUM CANDIDATES 4

text: 2.2 SIZE LIMITATIONS IN TYPICAL INTERNET PROTOCOLS 5

text: 2.3 PROTOCOLS ADAPTED TO POST-QUANTUM CRYPTOGRAPHY AND PERFORMANCE STUDIES 12

text: 2.4 SUMMARY 14

text: 3. NEW PROTOCOLS DESIGNED AROUND POST-QUANTUM 15 SYSTEMS

text: 3.1 USING KEMS IN PLACE OF SIGNATURES 15

text: 3.2 NEW DESIGNS TO DEAL WITH KEY SIZE 15

text: 3.3 NEW DESIGNS ADDRESSING DIFFERENT LAYERS 16

text: 3.4 SUMMARY

text: 17 4. DOUBLE ENCRYPTION AND DOUBLE SIGNATURES 17

text: 4.1 REVIEWING DOUBLE CRYPTOSYSTEMS 18

text: 4.2 DETAILS OF DOUBLE ENCRYPTION 19

text: 4.3 DETAILS OF DOUBLE SIGNING 19

text: 4.4 PERFORMANCE 19

text: 4.5 LONG-TERM PERSPECTIVE 20

text: 4.6 SUMMARY

text: 21 5. SECURITY PROOFS IN THE PRESENCE OF QUANTUM ATTACKERS 21

text: 5.1 QUANTUM ACCESS 22

text: 5.2 NEW MODELS 23

text: 5.3 REVISITING PROOFS 24

text: 5.4 SUMMARY

text: ii POST QUANTUM CRYPTOGRAPHY Integration study October 2022 6. STANDARDISATION EFFORTS FOR 25 PROTOCOLS

text: 6.1 SUMMARY

text: 27 7. CONCLUSIONS 30 BIBLIOGRAPHY iii POST QUANTUM CRYPTOGRAPHY Integration study October 2022

text: EXECUTIVE SUMMARY

text: With this report ENISA seeks to give insight on post-standardisation challenges. As a follow-up to ENISA’s 2021 Post-Quantum Cryptography: Current state and quantum miti- 1 gation study , the new report elaborates on the topic to address the following points:

text: • Integrating post-quantum systems into existing protocols • New protocols designed around post-quantum systems • Double encryption and double signatures using post-quantum systems • Seccurity proofs in the presence of quantum attackers • Standardisation efforts for post-quantum enabled protocols

text: The 2021 study provided an overview of the current state of play on the standardisation 2 process of Post-Quantum Cryptography (PQC) . It introduced a framework for analysing existing PQC proposals, presented the five (5) main families of PQC algorithms3, and 4 the NIST Round 3 finalists for encryption and signature schemes . It also sketched two proposals that proactive system owners can implement right now – before a standard is published – in order to protect the confidentiality of their data against a quantum capable attacker .

text: While agreeing on PQC cryptoalgorithms for encryption and signing is an important mile- 6 stone , by itself it is not enough. Any new cryptoalgorithm will need to interplay with existing protocols or even require entirely new protocols to be designed and implemented. Furthermore, PQC proposals are a solution to a still unrealised vulnerability – there are currently no publicly known quantum computers, strong enough to break encryption, 7 and not all scientists believe this will ever be the case –. Whether we should implement protections against a threat that might not materialise would be a moot question if said implementations were cost free. However, PQC algorithms are often more costly, e.g. in terms of size and computations. In addition, changing to a new cryptographic paradigm might provide new opportunities for software bugs and our understanding of the security 8 of the PQC algorithms is often less mature .

text: For each of the above open issues an overview of current developments is provided, along with future directions and identified gaps. Chapter 2 Integrating post-quantum systems into existing protocols provides an overview of the work done to integrate PQC proposals with current systems. It comments on the size and speed characteristics of the proposals, based on the benchmarks of the eBACS: ECRYPT Benchmarking of Cryptographic Systems 9 project [11], and how they interplay with the Internet Protocol (IP) and security protocols

text: iv POST QUANTUM CRYPTOGRAPHY Integration study October 2022

text: like TLS 1.3, VPN etc. As can be seen, not all use cases are created equal. For instance, in high-load cases, such as car2car communications, even apparently small differences between PQC proposals could have a significant impact by, for example, introducing latency or even incompatibility with existing communication protocols due to limits on message size. System designers will be required to understand the available options and make optimal choices for each use case, through calculated trade-offs.

text: A different approach would be to develop new protocols, taking into account the specifications of PQC systems from the design phase. The somewhat limited work done so far is mentioned in chapter 3 New protocols designed around post-quantum systems. The main outcome here is that existing work is promising but more research and deployment work is needed.

text: Chapter 4 Double encryption and double signatures takes on the veridical paradox that by striving for quantum resistance using a PQC system we might be lowering security overall. Actually, there is no guarantee that the post-quantum cryptosystems that survive the standardisation process are secure. So far cryptanalysts could have missed an important attack, perhaps even one that runs sufficiently quick on today’s non-quantum computers. Furthermore, the complicated new ecosystem of post-quantum cryptographic software has a clear risk of introducing bugs. A solution to this might be to augment, instead of simply replacing, current modern cryptosystems with PQC systems. This can be done by adding an extra layer that also encrypts and/or signs using post-quantum cryptography, as already discussed in our 2021 study. Here we take a closer look at the details and caveats of such a construction. The take away is that if this is done properly, then any attack will require breaking the current cryptosystem (e.g. one based on ellipticcurves) and breaking the post-quantum system. So, even if there are vulnerabilities in the post-quantum cryptosystem or post-quantum software, there will be no damage to the security of the existing system. The perceptive reader will have guessed that once more further investigations are required, but standardisation bodies are already working on this, specifying suitable mechanisms.

text: Chapter 5 Security proofs in the presence of quantum attackers, deals with formal models and proofs an important part of the analysis of modern cryptographic systems. While we have known for decades Shor’s and Grover’s algorithms – the former breaking RSA and ECC public key cryptography and the latter reducing the security level of symmetric cryptography – ongoing research on quantum computing might yet reveal more attacks against schemes and protocols. This is why cryptologists are not only working on proofs for the new PQC systems and protocols, but are also revisiting existing proofs for widely used systems and protocols. When we aim for post-quantum security, i.e. security against adversaries making use of a quantum computer, we have to model the adversaries also as quantum algorithms. This requires changing models and deciding about the specific abilities of quantum adversaries. New proofs have to be written that take quantum adversaries into account. This process has been started and is progressing well for basic building blocks, especially those considered in the NIST competition. However, in many other areas, especially in the analysis of protocols, the process has not even begun.

text: Finally, chapter 6 Standardisation efforts for protocols briefly discusses the work done by standardisation bodies, going beyond NIST’s seminal work, including ETSI, IETF and ISO, as well as recent reports by other European agencies, namely ANSSI and BSI. It is of interest to note that standardisation bodies continue to standardise protocols built using pre-quantum systems that will not withstand quantum attacks. In cases where significant developing investment has already been spent, one should consider applying the discussed concepts of double encryption, double signatures, etc. Otherwise the sensible thing is to consider post-quantum integration from the beginning when developing new standards.

text: 1 POST QUANTUM CRYPTOGRAPHY Integration study October 2022

text: 1 INTRODUCTION

text: Cryptography is a crucial tool for the security of our digital society and is used virtually everywhere. For example, it secures our online communications, keeps the data on our devices secret even if we lose them, and protects the integrity and authenticity of digital records. The security of cryptographic solutions deployed today is threatened by the development of quantum computers. To counter this threat, the area of post-quantum cryptography was initiated. Post-quantum cryptography studies cryptosystems under the assumption that the attacker has access to a quantum computer, while the user is supposed to be a regular user of today’s systems with no quantum capabilities. Several classes of problems exist that are conjectured to withstand even attacks using quantum computers. At this point the US National Institute for Standards and Technology (NIST) is running a selection process to select such systems for standardisation. Given the success of NIST standards in the area of cryptography, it is likely that the systems selected by NIST will become the international standard for large parts of the world, including the European Union. An overview of the process and the systems under consideration can be found in the recent Post-Quantum Cryptography: Current state and quantum mitigation 1 study by ENISA .

text: In July 2022, NIST announced four candidates to be standardised, plus the four fourth 2 round candidate Key-Establishment Mechanisms (KEMs) . In addition, NIST plans to issue a new Call for Proposals for public-key (quantum-resistant) digital signature algorithms by the end of summer 2022.

text: One might expect that with the end of this process, i.e. the publication of the standards, everything will have been solved. We simply replace the schemes that we are using today with the new systems and be done. This unfortunately is not the case at all. Many challenges need to be overcome before our data and our systems are secured against attacks using quantum computers.

text: One challenge is the size of the artifacts produced by post-quantum cryptography. The reason that today’s cryptographic systems are so efficient is also the reason why they are vulnerable to quantum computers – these problems are highly structured. Systems secure against quantum attacks have far less structure and hence a less compact description. As a consequence, keys, ciphertexts and signatures are larger for post-quantum systems than for matching pre-quantum systems; see Chapter 2. This poses challenges to higher-level protocols. Protocol messages do not meet the size limitations of underlying protocols anymore, which at least leads to fragmentation, additional round-trips, and a more complicated state-machine if not treated carefully. As a consequence, latency and data traffic increase.

text: A related aspect is a decrease in the speed of some algorithms. This can significantly hurt the performance of protocols if not taken into consideration. Both aspects can be dealt with by designing new protocols that take the new performance characteristics into account. However, this comes of course with all the challenges of designing new cryptographic protocols from assessing their security to standardising them.

text: A last aspect that prevents the use of the new cryptographic systems as plug-in replacements is that the encryption systems have a different interface. The currently most widely used cryptographic tool to establish a shared secret is the Diffie-Hellman key exchange. This is the Swiss Army knife of cryptography. Again, what makes this system so versatile is what makes it vulnerable to quantum attacks: its structure. None of the candidates in the NIST competition matches the data flow and versatility of the Diffie-Hellman key ex-

text: 2 POST QUANTUM CRYPTOGRAPHY Integration study October 2022

text: change. Hence, protocols that currently use the Diffie-Hellman key exchange have to be changed to interface with the new cryptographic systems.

text: Aside from the challenges of adopting and deploying post-quantum systems, analysing their security requires a novel set of tools and techniques to take attackers equipped with quantum computers into account. This has far-reaching consequences for establishing confidence that new systems are safe to deploy and in what circumstances. New security models have to be defined, new proof techniques have to be developed, and tools such as automated theorem provers have to be changed to be applicable in this new setting. Constructions that cannot be proven secure need to be replaced.

text: Finally, especially in the short term, many institutions will require modes that combine conventional and post-quantum cryptographic solutions to meet compliance regulations. The risk of current post-quantum proposals being broken is high enough that such combinations are recommended even when they are not required. This of course amplifies the performance issues mentioned above, as two systems have to be used in place of one. Dedicated combined modes can potentially reduce the cost but they have to be developed and studied.

text: Of course, a practical overhead that applies to all developed solutions is that they have to be standardised and implemented. This causes a whole chain of standards to be updated that make use of the new cryptographic systems or the protocols developed on top of it.

text: This study discusses the state-of-the-art with regard to these challenges. It provides the existing answers and highlights where more work is necessary.

text: 3 POST QUANTUM CRYPTOGRAPHY Integration study October 2022

text: 2 INTEGRATING POST-QUANTUM SYSTEMS INTO EXISTING PROTOCOLS

text: This chapter reports on work to use post-quantum systems in current protocols. It first presents data regarding size and performance of the NIST candidates, then reports on limits to sizes in Internet protocols and related experiments, and finally covers some case studies experimenting with integrating post-quantum cryptography.

text: For short explanations on the general functioning of these systems as well as details on the NIST round-3 candidates see the ENISA PQC study [13]. For more details see the chapters in standing document 8 (SD8) of ISO/IEC JTC 1/SC 27/WG 2 [25]. One of the authors, professor Lange, has developed an online course on post-quantum cryptography [49].

text: 2.1 SIZE AND SPEED OF POST-QUANTUM CANDIDATES

text: It is often said that post-quantum systems are larger or slower than their pre-quantum counterparts. Elliptic-curve cryptography (ECC) is certainly remarkably compact, achieving 128-bit security with just 256-bit keys. This compactness along with efficient implementations is seeing ECC replace the older RSA system since the mid-2000s when cryptanalysts had become more comfortable with the security analysis, security requirements had generally moved from 80 bits to 96 or 128 bits, and cryptographers had improved the speed of implementations to show significant advantages over RSA. For comparison, RSA keys at the 128-bit security level have 3072 bits. Key sizes for systems using discrete logarithms in the multiplicative group of finite fields are similar to RSA key sizes.

text: For post-quantum systems there is no overall winner. For example, for post-quantum keyencapsulation mechanisms (KEMs), the smallest key sizes are again achieved by systems related to elliptic curves, namely those based on isogenies, while the shortest ciphertexts are achieved by systems based on error-correcting codes. While isogeny-based systems also have relatively small ciphertexts and thus are small overall, code-based systems have some of the largest sizes for public keys, while lattice-based systems have medium sizes for ciphertexts and keys. When it comes to speed, isogeny-based systems are the slowest while systems based on lattices or codes are much faster.

text: A similar picture appears for post-quantum signatures. Multivariate quadratic systems have the smallest signatures, but by far the largest public keys, while lattice-based systems have medium-sized keys and signatures. Hash-based signatures have small keys but signatures a bit bigger than lattice-based signatures. They come with fast verification and 1 a bit slower signing speeds. In general, the size increase compared to ECC and RSA is worse for signatures than for KEMs.

text: This means that in the world of post-quantum cryptography, protocol designers need to be aware of the possibility of different trade-offs and choose systems matching their application scenario, taking into account (1) how frequently public keys are sent relative to ciphertexts or signed messages using them and (2) how important computation speed is relative to bandwidth.

text: The following sections present detailed benchmarking information for post-quantum KEMs and for post-quantum signatures.

text: 4 POST QUANTUM CRYPTOGRAPHY Integration study October 2022 2.1.1 Benchmarking data for KEMs

text: Figure 2.1 visualises these trade-offs by plotting the space needed for a ciphertext against the space needed for the public key for all NIST Round 2 KEMs which are submitted to eBATS [11] for benchmarking. Note the code-based system Classic McEliece [2] far out on the bottom right, showing the smallest ciphertexts and the largest public keys among the systems. The smallest total size of one ciphertext and one public key, towards the bottom left in the figure, is from the isogeny-based system SIKE [43].

text: For size measurements the CPU of the benchmarking system does not matter, but for the subsequent graphs, showing speed measurements, it does. We have chosen to include results measured on an Intel Haswell CPU as that was designated by NIST as the benchmarking platform for their post-quantum project. For the most up-to-date measurements on the same machine see https://bench.cr.yp.to/results-kem.html#amd64-hiphop.

text: Figure 2.2 plots the ciphertext size against the time taken for decapsulating a KEM ciphertext. Lattice-based systems such as Threebears [37], Kyber [61], NTRU [24] and NTRU Prime [9] show the fastest decapsulation speeds (smallest x-values) but Classic McEliece features the smallest ciphertexts (smallest y-values) while being only marginally slower than the lattice-based candidates. SIKE also has smaller ciphertext sizes (y-values) than the lattice-based systems but has much slower decapsulation speeds (x-values) placing it on the right edge of the figure.

text: The last graph we present for KEMs plots the public-key size against the time taken to generate a public key. Figure 2.3 shows that some systems are optimised for one-time key usage with fast key-generation speed while others are more suitable for long-term or few-time usage. Note that the term ‘ephemeral key’ does not mean one-time key but rather expresses that it must not exist anymore after some time period has passed, hence IND-CCA2 security is relevant for these systems; see [13] for definitions.

text: 2.1.2 Benchmarking data for signatures

text: Post-quantum signatures paint a similarly interesting picture when it comes to possible trade-offs. Figure 2.4 plots signature size, more precisely the space overhead for signed messages over the message size for signing a long message, against public-key size. This places systems roughly on the anti-diagonal, showing small public keys and larger signatures for hash-based signatures, such as SPHINCS+ [41], versus large public keys and small signatures for systems based on multivariate quadratics such as Rainbow [29] and GeMSS [23]. The lattice-based designs Falcon [58] and Dilithium [52] have medium-sized keys and signatures.

text: Signatures are typically verified multiple times, hence verification speeds matter. Figure 2.5 plots signature size, again meaning the space overhead for signed messages over the message size for signing a long message, against the time to generate a message given a signed message. Note that eBATS and NIST use the signed-message API, meaning that the verification algorithm takes a signed message as input and outputs the message or ‘failure’ depending on whether the signed message is valid or not, hence the verification step generates the message. Figure 2.6 plots public-key size against the time to generate a message given a signed message. These graphs show that designers will prefer different systems depending on whether public keys, i.e. certificates, are included or not. If public keys are included then systems based on lattices or hash functions have advantages over systems based on multivariate equations. The latter have very large public keys, making them attractive if public keys are rarely transmitted.

text: 2.2 SIZE LIMITATIONS IN TYPICAL INTERNET PROTOCOLS

text: The Internet transmits data inside IP packets. Each packet is individually addressed and has a limited size, so intermediate Internet routers can quickly receive a packet, forward the packet, forget the packet, and then reuse the same packet-handling resources for the next packet.

text: The latest version of the Internet protocol, IPv6, requires every router to be able to handle packets as large as 1280 bytes. The previous version, IPv4, is widely deployed and in theory does not guarantee that 1280 bytes are safe, but 1280-byte packets appear to be delivered throughout the Internet today without trouble. Larger IP packets are not prohibited but require the complications of ‘Path MTU discovery’ or the fragility of ‘frag-

text: 5 POST QUANTUM CRYPTOGRAPHY October 2022 Figure 2.1: Horizontal axis: space in bytes for a KEM public key. Vertical axis: space in bytes for a KEM ciphertext. Figure taken from https://bench.cr.yp.to/graph/ amd64-hiphop-kem-pkbytes,cbytes-nistpqc.pdf 6 POST QUANTUM CRYPTOGRAPHY October 2022 Figure 2.2: Horizontal axis: time in cycles to to generate a session key given a ciphertext. Vertical axis: space in bytes for a ciphertext. Figure taken from https://bench.cr.yp.to/graph/ amd64-hiphop-kem-kcycles,cbytes-nistpqc.pdf. 7 POST QUANTUM CRYPTOGRAPHY October 2022 Figure 2.3: Horizontal axis: time in cycles to generate a KEM public key. Vertical axis: space in bytes for a KEM public key. Figure taken from https://bench.cr.yp.to/graph/amd64-hiphop-kem-pkcycles, pkbytes-nistpqc.pdf. 8 POST QUANTUM CRYPTOGRAPHY October 2022 Figure 2.4: Horizontal axis: space in bytes for a public key. Vertical axis: space overhead in bytes for signing a long message. Figure taken from https://bench.cr.yp.to/graph/amd64-hiphop-sign-pkbytes, sbytes-nistpqc.pdf. 9 POST QUANTUM CRYPTOGRAPHY October 2022 Figure 2.5: Horizontal axis: time generate a message given a signed message. Vertical axis: space overhead in bytes for signing a long message. Figure taken from https//bench.cr.yp.to/graph/ amd64-hiphop-sign-mcycles,sbytes-nistpqc.pdf. 10 POST QUANTUM CRYPTOGRAPHY October 2022 Figure 2.6: Horizontal axis: time generate a message given a signed message. Vertical axis: space in bytes for a public key. Figure taken from https://bench.cr.yp.to/graph/amd64-hiphop-sign-mcycles, pkbytes-nistpqc.pdf. 11 POST QUANTUM CRYPTOGRAPHY October 2022

text: mentation’; see generally [20].

text: Many protocols, including HTTP and HTTPS, are built on top of TCP, which hides the packet-size limits by internally splitting the server’s stream of data into sufficiently small packets to send to the client. The client reassembles the packets into the original stream of data. The client’s stream of data to the server is similarly split into packets. TCP is capable of sending huge amounts of data such as long videos, much larger than the keys, ciphertexts and signatures in any of the post-quantum proposals under consideration.

text: However, some protocols are built directly on top of IP or on top of UDP, which is like IP in delivering only limited-size packets, and these protocols are not necessarily prepared for post-quantum sizes. Even for protocols built on top of TCP, sizes can still be a problem, because of limits in the protocols or in the software implementing the protocols. For example, Langley [50] found that the TLS software on various major web sites rejected ‘ClientHello messages larger than 3970 bytes’. These ClientHello messages are the simplest place to put post-quantum keys in TLS 1.3. Given this limitation, Google and Cloudflare discarded code-based systems and systems based on unstructured lattices from consideration for their joint post-quantum experiment CECPQ2 [48]. Cloudflare experimented with SIKE and NTRU-HRSS while Google also rejected SIKE, reportedly out of concerns over performance and denial-of-service attack potential; see, e.g. [47].

text: 2.3 PROTOCOLS ADAPTED TO POST-QUANTUM CRYPTOGRAPHY AND PERFORMANCE STUDIES

text: The works considered in this section fall roughly into three categories:

text: 1. Complete designs of protocols that offer security against quantum attacks and where the designers have made one choice of algorithm, which is motivated in the scientific paper. 2. Comparison studies analysing the suitability of different post-quantum systems for a protocol. These papers typically implement and analyse all potentially suitable systems and show the impact those choices have. They might give recommendations but typically include tables of results highlighting good choices for certain use cases. 3. Large Internet companies supporting post-quantum cryptography in prototype and deployments.

text: We will cover a selection of all categories.

text: 2.3.1 Post-quantum WireGuard

text: WireGuard [30] is a relatively new VPN protocol that breaks with the tradition of designing these protocols based on IPsec and instead builds them with modern systems and single choices, avoiding negotiation between participating nodes. While WireGuard in principle is designed to work between nodes that are equal, a typical deployment uses clients and servers and in general the protocol is asymmetric in that one party, the server, is known by a long-term key while the client can be registered with the server using a long-term key. These long-term keys are exchanged during a registration phase and are never sent during a protocol run. Instead, each party makes an ephemeral key for each connection and uses a combination of the resulting shared keys (ephemeral and long-term) for computing the session key which is then used to encrypt the content of the communication.

text: The post-quantum WireGuard design [42] uses the Classic McEliece system for the longterm keys and Saber for ephemeral keys. Their design benefits from the short ciphertexts provided by Classic McEliece and the fast key-generation speeds and compact key and ciphertext size of Saber, without encountering the cost for sending the large Classic McEliece keys. This allows the protocol to keep UDP packet sizes within the 1280-byte limit.

text: The publication shows a full implementation as well as a security proof for the new protocol.

text: 2.3.2 Web browsing secured with NTRU Prime

text: OpenSSLNTRU [10] is similar to CECPQ2 in that it integrates post-quantum key exchange into TLS 1.3 using a new one-time KEM key for each TLS session. The post-quantum details are different, achieving higher security with higher performance. Specifically, with

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text: the latest software, CECPQ2’s ntruhrss701 takes 359076 cycles on an Intel Haswell core for key generation plus encapsulation plus decapsulation, the total work of the client and the server; the key and ciphertext together use 2276 bytes. With the software from [10], OpenSSLNTRU’s sntrup761 takes 259472 cycles on an Intel Haswell core for key generation plus encapsulation plus decapsulation; the key and ciphertext together use 2197 bytes.

text: Furthermore, OpenSSH has supported Streamlined NTRU Prime since version 8.0 in 2019 2 and has it enabled on servers by default since version 9.0 that appeared in 2022. This means that clients will be free to select it without any action by the server administrator.

text: 2.3.3 Use of post-quantum cryptography for VPNs

text: In 2018, ETSI published [59] an analysis and comparison of the use of post-quantum systems in VPNs.

text: The document gives detailed descriptions of the Internet Key Exchange (IKE) protocol, the typical basis for VPNs built on IPsec, Transport Layer Security (TLS), used for transportation, the Media Access Control Security (MACsec) protocol, sometimes used for securing communications for the last mile between router and client, and finally the Secure Shell (SSH) protocol. For each of these protocols, requirements for drop-in replacements as well as for combined (hybrid) systems are analysed and recommendations are given. At 35 pages that study is too detailed to summarise here and the reader is invited to consult that source directly which is freely available.

text: 2.3.4 Studies analysing post-quantum systems for TLS

text: Several publications have tackled the TLS protocol. The Open Quantum Safe library (liboqs) [66] has made prototyping very easy by providing systems fitted into a fork of OpenSSL. The benchmarking of different systems for TLS is covered in [57] while combinations of different systems (hybrids) for key exchange is covered in [26] (see also Chapter 4). The latest TLS version is 1.3; [63] investigates post-quantum systems for authentication. Cloudflare very recently released a study [71] on the impact of signature and certificate sizes of post-quantum systems on TLS. This study also addressed problems regarding authentication.

text: 2.3.5 Signature verification on very small devices

text: Some processors are so small that they cannot hold the full public key or signature for 3 some of the post-quantum systems. Signature verification in feature activation in cars is such a scenario and is the topic of [35]. The paper analyses the impact on performance when the RAM size is restricted to 8kB, meaning that signatures and/or keys need to be streamed into the device. Should the public key need to be streamed in, the device needs to have a hash of the public key in secure memory in order to compare the streamed-in key against this hash. For the large-key systems Rainbow and GeMSS this caused a serious impact on performance due to the extra hashing. SPHINCS+ naturally has a very small public key and the implementation ran with only a few adjustments on the ARM Cortex- M3 development board. For Dilithium the fastest implementation required too much code size, while key size and signature size posed no problems.

text: 2.3.6 Signature verification and generation in a high-load scenario

text: Car-to-car communication needs to be authenticated to prevent rogue stations from creating accidents by causing cars break suddenly to avoid colliding with non-existing obstacles. On a busy street, cars receive many signatures per second and need to check the validity and react on the contents without losing time. Cars also continuously send messages for which they need to generate signatures.

text: A recent study [17], presented at NIST’s round 3 workshop, showed that the differences

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text: between Dilithium and Falcon, which seem rather small in the benchmarks presented in Section 2.1, have a significant impact in aggregation and that the latency of signature transmission, i.e. the size overhead, makes a significant difference in car-to-car communication.

text: 2.3.7 Large companies supporting post-quantum Internet solutions

text: Amazon Web services advertises support for post-quantum TLS for key-management services [38]. Similarly, Microsoft now supports post-quantum cryptography for VPN connections [31].

text: 2.4 SUMMARY

text: Post-quantum systems have different trade-offs in size and speed than their pre-quantum counterparts. Protocol designers need to understand the space of options presented in Section 2.1 to make optimal choices for each use case. This chapter presents the impact on performance of these choices. Chapter 5 covers the implications for security proofs.

text: Six example studies are presented to show the options for adding post-quantum protection to existing protocols. Making these choices, in particular if bandwidth is an issue, is still a topic of academic papers and some expertise is needed. The graphs and summaries in Section 2.1 can provide guidance.

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text: 3 NEW PROTOCOLS DESIGNED AROUND POST-QUANTUM SYSTEMS

text: A different approach to dealing with the difference in interfaces is to design new protocols which are developed around post-quantum systems, taking their specifications into account. This is likely the option that leads to the most efficient protocols but it requires a lot of work. The number of published proposals is still somewhat limited and more work in this direction is required.

text: 3.1 USING KEMS IN PLACE OF SIGNATURES

text: Already in a pre-quantum world, Diffie-Hellman and KEMs are more efficient than signatures, making it attractive to design protocols using these primitives for authenticity and secrecy rather than using them only for secrecy and using signatures for authenticity. In the Diffie-Hellman context this can be traced back to at least 2006 when Bernstein pre- 1 sented very efficient DH speeds and proposed to use DH in place of signatures.

text: In the post-quantum context, and in particular for lattices, signatures are both larger and less efficient to compute than KEMs, which prompted [28] to investigate the security of combining two KEMs, one with ephemeral keys for forward secrecy and one with longterm keys for authentication. The paper lays out the combinations and gives a proof in the random oracle model. This approach is also taken in PQWireGuard described above.

text: For TLS applications, where typically one side is authenticated, [62] describes a combination of two post-quantum KEMs and reports implementation results.

text: 3.2 NEW DESIGNS TO DEAL WITH KEY SIZE

text: The public keys in the code-based system Classic McEliece are 1 MB and larger for the 5th category. This had led to recommendations to use the system only for long-term keys, as using them ephemerally would cause a high level of fragmentation. The McTiny protocol [12] deals with the scenario in which a client transmits an ephemeral key to the server while the server has a long-term Classic McEliece key used for authentication and to protect the integrity and secrecy of the key shares and partial encryptions. The protocol uses the intrinsic properties of code-based systems to handle encryptions using parts of the key. The paper considers the denial-of-service attack surface posed to servers accepting large keys and presents a design in which servers do not allocate any per-client memory beyond the size of an incoming internet packet and fully avoid memory-flooding DoS. The protocol is fully implemented and tested for latency and deploys congestion control.

text: 3.3 NEW DESIGNS ADDRESSING DIFFERENT LAYERS

text: Besides interface and speed issues, a concern with replacing pre-quantum systems with post-quantum systems is that it is a slow process which requires years of discussions in the standardisation bodies that control the systems. Even if NIST’s choices will be adopted without changes and those bodies start working even before the full specifications of the new standards are announced, several more years of data will be protected purely with pre-quantum systems.

text: An avenue pursued in parallel is to leverage the layer structure of the Internet to add post-quantum protection for all connections between nodes that offer support. PQConnect [8, 51] builds secure tunnels between a user’s browser and supporting websites.

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text: Each connection is secured with two long-term identity keys, one using the code-based Classic McEliece system, the other using pre-quantum elliptic-curve cryptography, and two ephemeral keys, combining elliptic curves with structured lattices. The combination is done in such a way that the (potentially) weaker systems are wrapped inside the stronger ones, forcing attackers to break the stronger systems to gain access to those ciphertexts and keys. The existing prototype uses a SOCKS proxy but the finished project should operate in kernel space the same way that a VPN does and protect the full communication from DNS lookups to data transfer.

text: 3.4 SUMMARY

text: To tackle the challenge of transitioning to post-quantum cryptography some new protocols have chosen to break out from replacing (or augmenting; see Chapter 4) prequantum systems with post-quantum systems of the same type. These protocols are instead designed around post-quantum systems. Examples covered are replacing signatures with KEMs, putting key-sizes as guiding factors, and offering post-quantum protection at a different layer of the Internet. The latter has the additional benefit of promoting a fast rollout, not hampered by slow standardisation bodies.

text: So far only a few studies have departed from the standard approach of replacing prequantum with post-quantum or strapping on post-quantum systems, but existing works are promising and more research and deployment work is needed in this area.

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text: 4 DOUBLE ENCRYPTION AND DOUBLE SIGNATURES

text: Almost all of the post-quantum cryptosystems submitted to NIST have lower security against the best attacks known today than against the best attacks that were known in 2017. In many cases the loss of security was severe enough that the submissions are categorised as broken. There is no guarantee that the surviving post-quantum cryptosystems are secure. Cryptanalysts could have missed an important attack, perhaps even an attack that runs quickly on today’s computers. Furthermore, many serious bugs have already been discovered in the complicated new ecosystem of post-quantum cryptographic software and there is a clear risk of further bugs. State-of-the-art tools can guarantee that cryptographic software does not have buffer overflows, but formal verification of full functional correctness remains challenging.

text: Consequently, there is a risk that switching from pre-quantum elliptic-curve cryptography to a post-quantum cryptosystem will damage security, not just failing to protect against quantum computers but also losing protection against today’s computers.

text: To address this risk, you should deploy post-quantum cryptography today as an extra layer together with pre-quantum cryptography, rather than deploying it as a replacement for pre-quantum cryptography. Sign with a pre-quantum cryptosystem and with a postquantum cryptosystem, making sure that the verifier checks both signatures. Encrypt as usual with pre-quantum cryptography, and then encrypt the result with post-quantum cryptography, so that decryption requires both secret keys.

text: For example, Google’s ‘CECPQ1 (combined elliptic curve + post-quantum)’ key-exchange mechanism [21] combines a well-known pre-quantum elliptic-curve cryptosystem, X25519, with a post-quantum lattice-based cryptosystem, NewHope-1024:

text: The post-quantum algorithm might turn out to be breakable even with today’s computers, in which case the elliptic-curve algorithm will still provide the best security that today’s technology can offer.

text: CECPQ2a combines X25519 with another post-quantum lattice-based cryptosystem, NTRU-HRSS-701. CECPQ2b combines X25519 with a post-quantum isogeny-based cryptosystem, SIKE-p434. In each case, even if the post-quantum cryptosystem completely fails, the user’s data is protected by X25519. Large quantum computers will eventually break X25519, but this failure is farther in the future than the immediate disaster that would occur if X25519 were omitted.

text: 4.1 REVIEWING DOUBLE CRYPTOSYSTEMS

text: Double encryption and double signing sound straightforward, but one must be careful with the details. For example, it is dangerous to reuse keys across multiple systems. It is dangerous to reuse any randomness used for encryption, signing, etc. It is dangerous to assume that the presence of one component will compensate for weaknesses in another component: 1993 Maurer–Massey [53, Section 1] gave an example where inserting one weak encryption layer exposes a sample application to the weaknesses of a second encryption layer, where the second encryption layer by itself would not have been exploitable for the same application. Each component should be designed to be secure by itself.

text: A well-designed double cryptosystem makes it easy for security auditors to see that the cryptosystem is strong if at least one of the component cryptosystems is strong. This means, in particular, that adding post-quantum cryptography does not damage the prequantum security provided today by elliptic-curve cryptography. This also provides a

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text: straightforward way to integrate post-quantum cryptography while remaining in compliance with existing regulations and policies that require elliptic-curve cryptography. Double cryptosystems thus help fast deployment of post-quantum cryptography.

text: Names in the literature for the same concept include ‘double encryption’ (and, more generally, ‘multiple encryption’), ‘combiners’, and ‘hybrid cryptosystems’. Be aware that the name ‘hybrid cryptosystem’ is also used for various constructions requiring all of the component cryptosystems to be strong: the standard example is encrypting data using a cipher key that was exchanged with a public-key cryptosystem.

text: Security analysis of the CECPQ1_RSA_WITH_AES_256_GCM_SHA384 TLS cipher suite starts from the assumption that CECPQ1 encryption and RSA signatures and the AES-256-GCM authenticated cipher and SHA-384 are secure; security analysis specifically of CECPQ1 starts from the assumption that X25519 or NewHope is secure.

text: ETSI [60] has specified mechanisms to combine pre-quantum encryption with postquantum encryption. Another specification [65] is under development within IETF. NIST’s key-derivation standard [6, Section 2, second paragraph] permits a standard shared secret (which for NIST currently means a pre-quantum shared secret) to be concatenated with an ‘auxiliary shared secret’ (which NIST has not specified how to obtain, but would presumably be a post-quantum shared secret), with the concatenation hashed in specified ways to obtain a key.

text: 4.2 DETAILS OF DOUBLE ENCRYPTION

text: Recall that user data is normally encrypted and authenticated with an authenticated cipher such as AES-256-GCM or ChaCha20-Poly1305, after an elliptic-curve key-exchange mechanism is used to establish the 256-bit session key used by this authenticated cipher.

text: These authenticated ciphers are not threatened by quantum computers. Shor’s algorithm does not apply to them, and the key sizes are large enough to leave a comfortable security margin even considering Grover’s algorithm. See, e.g. [4]. The possibility of double encryption using, e.g. AES-256-GCM together with ChaCha20-Poly1305 is outside the scope of this document.

text: What is threatened by quantum computers is the elliptic-curve key-exchange mechanism. For example, consider the ECDH KEM: there is a standard elliptic-curve point 𝐺; Alice generates a secret key 𝑎 and sends a public key 𝐴 = 𝑎𝐺; Bob generates a secret 𝑏 and sends a ciphertext 𝐵 = 𝑏𝐺; Alice and Bob now compute the same point 𝑎(𝑏𝐺) = 𝑏(𝑎𝐺), and hash this point to obtain a session key 𝑘. Assume that all of the objects 𝐴, 𝐵, 𝑘 are encoded as fixed-length strings.

text: 𝐴′ Now consider a post-quantum KEM where Alice generates a fixed-length public key ; 𝐵′ Bob generates a fixed-length ciphertext ; Alice and Bob compute a fixed-length ses𝑘′ sion key . You should build a double KEM as follows, with keys and other random objects generated independently for the ECDH KEM and the post-quantum KEM:

text: 𝐴, 𝐴′ • Alice’s public key in the double KEM is the concatenation ( ); 𝐵, 𝐵′ • Bob’s ciphertext in the double KEM is the concatenation ( ); 𝐵, 𝐵′ • The session key encapsulated under public key ( ) is a standard hash of the concate𝑘, 𝑘′, 𝐵, 𝐵′ nation ( ).

text: 𝐴, 𝐵, 𝑘, 𝑘′ These concatenations are unambiguous since have fixed length.

text: This description implicitly assumed that the two input KEMs (the ECDH KEM and the postquantum KEM) are ‘quiet’, always returning session keys, never rejecting ciphertexts. To 𝐵, 𝐵′ allow non-quiet KEMs, define the double KEM to reject ( ) if the first input KEM rejects 𝐵 𝐵′ or the second input KEM rejects .

text: The above construction is an example of the ‘first proposal’ in [34, Section 1.1]. See [34] and [14] for further constructions; these papers identify the necessary security properties 𝑘, 𝑘′, 𝐵, 𝐵′ achieved by the hashing of ( ), and use this to replace the hashing with something faster. See also [40] for an even faster construction that merges this hashing with the two hashing layers inside the two KEMs, although this no longer treats the KEMs as black boxes. Finally, see [65] for a variant that omits this hashing in the context of TLS (which has a subsequent hashing layer) and for an analysis of options for negotiating key-

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text: exchange choices.

text: 4.3 DETAILS OF DOUBLE SIGNING

text: There are two common interfaces for signature systems:

text: • A ‘detached signature’ is separate from the message. Verification takes a message and a detached signature, and returns acceptance or rejection. • A ‘signed message’ (also called a ‘signature with message recovery’) includes the message. Verification takes a signed message and returns a message or rejection.

text: The signed-message interface avoids the risk of forgetting to verify a signature. Sometimes signed messages are designed to take less space than a message plus a separate signature. One can generically convert a signed-message interface into a detachedsignature interface by signing a hash of the message, although this loses the benefits of a signed-message interface.

text: With a detached-signature interface, combining a pre-quantum signature system 𝑃 with a post-quantum signature system 𝑄 is a simple matter of concatenating the two signatures, assuming the signatures have fixed length. The double-signature verifier accepts the double signature exactly when the 𝑃 verifier accepts the 𝑃 signature and the 𝑄 verifier accepts the 𝑄 signature.

text: Beware that some proposed post-quantum signature systems have variable-length signatures. Concatenation is still safe as long as the first system has fixed-length signatures. An alternative is to use an unambiguous encoding of variable-length strings, as in the ‘composite’ signatures in [56]. Beware that [56] points to patent applications regarding multiple signatures; for prior art see, e.g. [16].

text: With a signed-message interface, combining a pre-quantum signature system 𝑃 with a post-quantum signature system 𝑄 is a simple matter of signing first with 𝑃 and then signing the result with 𝑄. The double-signature verifier applies the 𝑄 verifier followed by the 𝑃 verifier.

text: An alternative is to sign first with 𝑄 and then with 𝑃. Signing first with 𝑃 has the advantage that the auditor can entirely ignore 𝑄 in checking that any message produced as output was verified by 𝑃. If 𝑄 is first, then the 𝑃 auditor also has to check a minimal correctness property of 𝑄, namely that if a message is signed by 𝑄 and then given to the 𝑄 verifier without modification then the output of verification cannot be any other message.

text: There are more complicated double-signing proposals that modify how signatures are used in applications: for example, modifying TLS to check a certificate using a 𝑃 signature and to check a separate certificate using a 𝑄 signature. It is simpler to build an application-independent double-signature system and upgrade applications to use that system.

text: 4.4 PERFORMANCE

text: As explained earlier in this document, post-quantum cryptosystems send more data than elliptic-curve cryptosystems. For example, a NewHope-1024 ciphertext is 2208 bytes, an NTRU-HRSS-701 ciphertext is 1138 bytes, a compressed SIKE-p434 ciphertext is 236 bytes, and an X25519 ciphertext is just 32 bytes. Applications that can afford the data sent by post-quantum cryptography are likely to be able to afford an extra 32 bytes for double encryption with elliptic-curve cryptography and post-quantum cryptography. Similar comments apply to double signing.

text: However, hybrids can be a larger performance problem in other cost metrics. For example, there has been some investigation of compact implementations of post-quantum cryptography for small devices (see, for example, [72]), and there are various ideas on how to save space by merging implementations of post-quantum cryptography with implementations of elliptic-curve cryptography (see, for example, [5]). The details and costs require further investigation.

text: 4.5 LONG-TERM PERSPECTIVE

text: One hopes that security analysis of post-quantum cryptosystems will eventually stabilise, preferably without many more broken cryptosystems. Furthermore, the fact that elliptic-

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text: curve cryptography has security value today does not mean that it will have security value forever. The first public use of a quantum computer to compute a 256-bit elliptic-curve discrete logarithm will naturally raise questions regarding the continued value of deploying elliptic-curve cryptography, even if the break is very expensive. Assuming that there is public consensus someday that elliptic-curve cryptography is obsolete, it will then be reasonable to consider removing it from protocols such as TLS, with the goal of eventually allowing simpler TLS implementations.

text: 4.6 SUMMARY

text: Start with a system that encrypts and/or signs using elliptic-curve cryptography. Add an extra layer that also encrypts and/or signs using post-quantum cryptography. This is ‘double encryption’ and/or ‘double signing’. If this is done properly, then any attack will require breaking the elliptic-curve system and breaking the post-quantum system, so upgrading to post-quantum cryptography will not damage the [cryptographic] security of the existing system, even if there are vulnerabilities in the post-quantum cryptosystem or 1 post-quantum software . State-of-the-art tools can guarantee that cryptographic software does not have buffer overflows, but formal verification of full functional correctness remains challenging. In all cases, the details must be handled carefully for security. Usually there is no real performance reason to remove elliptic-curve cryptography.

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text: 5 SECURITY PROOFS IN THE PRESENCE OF QUANTUM ATTACKERS

text: The common expectation is that quantum computers only threaten the security of public key cryptography (PKC), more specifically of the basic PKC schemes, such as encryption, KEM, and signature schemes. However, this is mostly based on the already existing knowledge of attacks that threaten these schemes while attacks against other schemes and protocols are not known for now. A common approach in modern cryptography is to reduce the attack surface against the mathematical security of a protocol or system to break the security of its basic building blocks via security proofs. Building blocks in this case can either be mathematical problems such as the RSA problem or the LWE problem, or these can be higher level building blocks like KEMs, signature schemes, or hash func- 1 tions for which we already established security by another proof or cryptanalysis .

text: Clearly, if new protocols are constructed these require new proofs. But even existing proofs for well-known protocols have to be revisited. The reason is that when we aim for post-quantum security, i.e. security against adversaries making use of a quantum computer, we have to model our adversaries also as quantum algorithms. This requires changing models and deciding about the specific abilities of quantum adversaries. Afterwards, proofs have to be vetted again in the new setting. This process has been started and is progressing well for basic building blocks, especially those considered in the NIST competition. However, in many other areas, especially in the analysis of protocols, the process has not even begun. In this section we discuss known results and afterwards summarise open questions for all dimensions of this problem. For a more 2 detailed overview of recent topics in this area, see the talks of the Quiques workshop , co-organised by one of the authors, professor Hülsing and professor Majenz.

text: 5.1 QUANTUM ACCESS

text: Security games often model the attacker as having access to oracles. For example, in the standard definition of security for signatures, the adversary is given a public key and access to a signing oracle that allows it to learn signatures on messages of its choice under the corresponding secret key.

text: The first question when revisiting existing security models is whether an adversary should be given quantum access to the oracles provided and, if so, to which ones. In this context, quantum access to an oracle for a function 𝑓 refers to the ability to make a query with a superposition of inputs for 𝑓 and receive the corresponding superposition of outputs of 𝑓. More formally, a quantum superposition oracle for a function 𝑓 : 𝑋 → 𝑌 is defined as the unitary transform

text: ∑ ∑ 𝑈𝑓 : 𝛼𝑥, 𝑦|𝑥, 𝑦⟩ → 𝛼𝑥, 𝑦|𝑥, 𝑦 ⊕ 𝑓 (𝑥)⟩, 𝑥∈𝑋, 𝑦∈𝑌 𝑥∈𝑋, 𝑦∈𝑌 where ⊕ refers to bitwise xor. It should be noted that given a classical description of 𝑓 this quantum oracle can be implemented using standard techniques.

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text: Standard model vs idealised models. In this context, an important distinction has to be made. This is between the standard model and idealised models such as the random oracle or the ideal cipher model. In the standard model, the adversary is only given oracle access to functionalities that it cannot compute itself, usually because they use a secret key that is not available to the adversary as in the case of the signing oracle. In idealised models, some cryptographic building blocks are replaced with ideal versions, e.g. hash functions with random functions (random oracle model) or block ciphers with a collection of random permutations (ideal cipher model). These ideal building blocks do not allow for efficient descriptions. Hence they are introduced into the model as oracles to which all parties have access.

text: Three different classes of models. For decisions on which oracles should be quantumaccessible, the following classification, introduced in [32] (with prefix ‘QS’) and now widely used (with prefix ‘Q’), distinguishes the most relevant models:

text: • In Q0 models, access to all oracles is classical. • In Q1 models, quantum access is given to oracles that simulate functions from an ideal model that are not secretly keyed. Oracles implementing secretly keyed functions only allow for classical access. • In Q2 models, quantum access is given to all oracles, including those that implement secretly keyed functions.

text: The Q0 models cover the setting of a world where no quantum computers exist.

text: When we want to consider the setting in which an adversary has access to a quantum computer we have to consider Q1 models. The reason is that in idealised models, we are modelling functions that are publicly known as oracles. In the real world, these functions could be implemented as quantum algorithms by an adversary and then run on a quantum computer. Therefore, these functions are computable in superposition by the adversary in the real world. Q1 models reflect this by providing quantum access to ideal functionalities that are not secretly keyed. While it was questioned if this makes a relevant difference, this question has been recently settled in the affirmative by [73] giving a separation between the classical random oracle model (ROM) and the quantum-accessible random oracle model (QROM).

text: Q2 models go beyond this. In general, Q2 models would apply in a world where honest users work on quantum computers, connected via a quantum network with other users and the adversary. In this setting, it might happen that an adversary gets quantum access to secretly-keyed functionalities. Another case where one would have to apply the Q2 model is the exceptional case where an adversary is actually given quantum access to such secretly keyed functionalities although honest users still use classical computers. This is, for example, the case in a setup where the adversary is given an obfuscated circuit of a secretly-keyed block cipher. In this case, the adversary could translate the circuit into a quantum circuit and obtain quantum access.

text: In summary, when one aims for protecting the current infrastructure against an adversary that may have access to a quantum computer, Q1 models are generally the right choice. However, Q2 models can be relevant in the far future or for analyses of some advanced protocols.

text: 5.2 NEW MODELS

text: After deciding which oracles have to allow for quantum access the quantum counterpart of the old (classical) security model can be defined by changing the following two of its aspects: Firstly, the adversary is assumed to be a quantum algorithm. Secondly, the adversary is given quantum access to the selected oracles. This gives a valid model. However, it does not guarantee that the model is useful, as security in this model may be unachievable. This problem occurred often in Q2 models, necessitating the development of new security models (e.g. see [19, 33, 1]).

text: In this report, we are concerned primarily with Q1 models. For Q1 models, the standard model only changed by allowing the adversary to be a quantum algorithm. In this setting, post-quantum security is achievable for most tasks if we assume that mathematical problems that are hard for quantum computers to solve do exist. However, even in this case it turns out that models have to be carefully analysed. Some models may not reflect the

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text: intended behaviour anymore. For an example, see [68].

text: When it comes to idealised models, the most widely used model, the random oracle model (ROM), was already analysed in 2011 and the quantum-accessible random oracle model (QROM) was introduced as its Q1 counterpart [18]. So far, most security notions seem to be also achievable in the QROM. Its sibling, the quantum ideal cipher model, was only introduced in 2018 [39]. Known examples also pose achievable security notions.

text: The random oracle and ideal cipher model are the most basic idealised models. When it comes to the analysis of more advanced constructions or protocols, two common models are the indifferentiability framework [54] and the universal composability (UC) framework [22]. The general possibility of proofs of post-quantum security in the indifferentiability framework was recently demonstrated [74]. The general possibility of proofs in stronger variants of this framework is the subject of ongoing investigation. The achievability of proofs of post-quantum security in the UC model is still an open question. It is known that the UC model can be extended even to the Q2 setting when considering information-theoretic security; but there is an important distinction between informationtheoretic results and complexity-theoretic results. The UC framework is commonly used in the analysis of cryptographic protocols, and complexity-theoretic results are necessary for establishing the security of protocols that use public key cryptography.

text: 5.3 REVISITING PROOFS

text: When the models are fixed, a last challenge is to vet the existing proofs or to write new ones where necessary. Due to the focus of this report, we restrict ourselves in the following to results for the Q1 models. Proofs in the standard model do still apply when considering quantum adversaries as long as they avoid certain problematic techniques, most notably rewinding. A formal analysis that describes the precise conditions under which a classical proof directly translates to the corresponding Q1 model appeared in [64]. Unfortunately, these seem to be the only proofs that remain intact. Proofs that make use of the mentioned techniques have to be rewritten. While the question of rewinding proofs in a quantum setting has been the subject of a long line of research, many open questions remain (cf. [67] or the lecture by Unruh at Quiques [70]).

text: Proofs in idealised models generally have to be revisited as the oracles change. When the oracles become quantum-accessible a lot of commonly used proof techniques become inapplicable. Notable exceptions are so-called history-free reductions. For these it was shown in [18] that they also apply in the QROM.

text: The two main reasons for ROM proofs to fail in the QROM are as follows. Firstly, we have to measure a quantum register to learn its state. However, a measurement can and in most cases does change the state of the register. As a consequence, a reduction cannot learn the queries an adversary makes to its oracles anymore, which was a crucial step in many proofs. Secondly, an adversary may query an oracle at all possible inputs with a single query. This causes the failure of proofs that reason about an adversary not querying an oracle on a certain position, another technique that is frequently used in proofs. Both issues have been solved in recent works [74, 3, 36].

text: Generally, most of the results in the ROM have been recovered in the QROM though with less tight bounds. Most importantly, new security proofs in the QROM are known for the different constructions of KEM and signature schemes used for the NIST proposals (see Chapter 3 of the recent ENISA study Post-Quantum Cryptography: Current state and 3 quantum mitigation ). For KEMs that are constructed from a deterministic PKE scheme, the QROM proofs are almost as tight as the proofs in the ROM [15, 46]. However, these form a notable exception as most other proofs are significantly less tight than the corresponding proofs in the ROM. The problem with less tight bounds is that with such bounds parameter selection is not supported by the proofs as the relation between the security of a scheme and the underlying mathematical problems is too loose. This is especially dangerous as we are lacking experience when it comes to the impact of quantum attacks on schemes and protocols. Even if they do not fully break them, they can significantly decrease the security level (see for example the case of MQDSS, a second round NIST

text: 23 POST QUANTUM CRYPTOGRAPHY Integration study October 2022

text: candidate [44]). Hence, more research is necessary on new techniques that allow tighter proofs in the QROM.

text: The two main open questions when it comes to giving proofs of post-quantum security refer to the handling of random permutations and again the UC framework. So far, several attempts have been made to extend the solution that allows to observe adversarial queries in the QROM [74] to permutations [27, 69] but so far all of them have turned out to be flawed. This problem is relevant to the analysis of SHA-3. As mentioned above, so far the UC model is unexplored territory in the post-quantum setting. The UC model is necessary whenever different instances of a protocol or scheme may be composed in some way to build a bigger scheme. Hence, many protocols, including password-authenticated key exchange, and multi-party computation protocols, are commonly proven to be secure in the UC framework. Therefore, it is a necessity to explore this space in order to also recover security proofs for these protocols.

text: 5.4 SUMMARY

text: When considering adversaries equipped with quantum computers, not only the mathematical assumptions have to change but also the models and security proofs in use. In the foreseeable future, the relevant security models are Q1 models, meaning that adversaries get quantum access to oracles implementing public functions and classical access to oracles implementing secretly keyed functions (like a signing or a decryption oracle). The most prominent Q1 model is the quantum-accessible random oracle model (QROM). Proofs for basic building blocks, such as KEMs and signatures, have been recovered in the QROM. However, the bounds are worse. Thus, they often cannot support the claimed security for proposed parameters – except for a few notable exceptions as discussed above.

text: Beyond basic primitives, new models and proofs are still sparse. More research on new models and proof techniques for these models is required. This will enable us to justify parameter choices for post-quantum cryptography by security proofs.

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text: 6 STANDARDISATION EFFORTS FOR PROTOCOLS

text: Besides NIST, other standardisation bodies have also been busy with preparing for postquantum cryptography.

text: ETSI, the European Telecommunications Standards Institute, has established a working 1 group on Quantum-Safe Cryptography (QSC) . The working group has published several documents including one on protocols and standards adapted to post-quantum cryptog- 2 raphy .

text: The IETF (Internet Engineering Task Force) defines protocols for Internet communication. Different protocols are handled by different working groups. The CFRG ()Crypto-Forum Research Group) which handles cryptosystems for different applications, has deferred standardising any KEMs and signature schemes until after the end of the NIST process. However, other working groups have investigated how the protocols under their guard can be adapted. For example the IKE (Internet key exchange) working group has formulated how to use pre-shared keys in IKE in Mixing Preshared Keys in IKEv2 for Post- 3 quantum Security . The IKE protocol is used in IPsec for VPNs. There are also proposals to the TLS working group regarding mechanisms to use various NIST candidates in TLS, and 4 a proposal Quantum Safe Cryptography Key Information regarding key serialisation. The LAMPS working group is discussing an Internet draft for supporting PQ in certificates [76]. Further working groups started similar discussions but do not have Internet drafts yet.

text: ISO’s working group on cryptography, SC 27/JTC 1 WG 2, has conducted a two-year study period on post-quantum cryptography during which the experts were briefed by researchers in the field on the functioning of and main avenues for the families of postquantum cryptography. While discussions about choosing candidates have been delayed till after NIST’s decision, the working group has published a standing document [25] describing all families of systems.

text: Following a workshop on ‘Considerations in Migrating to Post-Quantum Cryptographic Algorithms’, NIST’s National Cybersecurity Center of Excellence (NCCoE) wrote a summary report [7], covering five application scenarios which currently deploy pre-quantum cryptography and outlining the high-level architecture and referencing relevant standards.

text: While we recognise NIST’s leading role in standardising cryptographic systems we recommend that all stakeholders – governments, industry, and data-protection officers as well as other standardisation bodies – acquire sufficient understanding of post-quantum cryptography to make informed decisions.

text: Most recently, after the draft for this report was finalised, the French cybersecurity agency ANSSI published its views on the transition to post-quantum cryptography [55]. Beyond giving common advice to use double encryption, but without going into the details, the report also cautions about the immaturity of post-quantum algorithms, saying that they are still in the research phase. The report does not make any recommendations regarding which systems to choose but states explicitly that systems which are not chosen by NIST but are demonstrably stronger also qualify for security evaluation and certification. Even more recently, a report by the German BSI appeared [75] discussing their view on postquantum security and giving recommendations for the next steps. Similar to ANSSI, the

text: 25 POST QUANTUM CRYPTOGRAPHY Integration study October 2022

text: BSI recommends the use of double encryption and signatures, and highlights the need for further research on post-quantum cryptography. Moreover, the BSI report provides background and makes recommendations for short-term protective measures.

text: The above-mentioned works show progress towards securing infrastructure against quantum attacks. At the same time it should be noted that standardisation bodies continue to standardise protocols built using pre-quantum systems that will not withstand quantum attacks. For example, the CFRG is currently in the process of selecting a new standard for password-authenticated key exchange (PAKE). It seems advisable to also apply the concepts of double encryption, double KEM, double signatures, etc. – see Chapter 4 – to such advanced protocols. However, this is not on the current roadmap. Worse, the existence of a recent standard can make it a lot harder to motivate an organisation or the community to develop a new standard. Therefore, we recommend that post-quantum integration should already be considered when developing new standards.

text: 6.1 SUMMARY

text: Several standardisation bodies have agendas for standardising post-quantum systems and protocols. However, not all new initiatives for standards consider security under quantum attacks and more attention to future-proofing is necessary. Furthermore, we recommend that all relevant parties acquire knowledge in post-quantum cryptography and not rely solely on external standards.

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text: 7 CONCLUSIONS

text: We had already discussed in ENISA’s 2021 ’Post-Quantum Cryptography: Current state and 1 quantum mitigation’ study that Quantum Technology and in particular Quantum Computing are set to be a major disruptor. We have known for more than two decades that the development of a sufficiently large andpractical quantum computing machine will render most cryptographic systems insecure, radically transforming the existing threat model and endangering our infrastructure. Current public key cryptosystems, used in ecommerce, digital signatures, electronic identities etc., will be the most impacted by such a development.

text: While such a system does not yet exist, there are several ongoing large scale investments from both industry players and nation states. However, not all development in the area is public and it is not unthinkable that the first fully functional large quantum computer will not be publicly announced. In addition, rolling out new cryptographic systems takes a lot of time and effort; it might even be infeasible for systemswith restricted accessibility, such as satellites. Thus, policy makers and system owners should make preparations.

text: Deciding on a new quantum-resistant cryptoalgorithm, like the ones selected by NIST, is only one part of the solution. Any PQC proposal will need to either integrate with existing systems and protocols or will require the design and deployment of new systems and protocols. Neither is a trivial task. The differences between existing cryptoalgorithms and PQC, in terms of key and ciphertext size, and computation times, even when small in absolute numbers, might in aggregate introduced unexpected problems for specialized use cases; such as in car2car communications. Which means that we might end up with multiple options, each catering for different usecases and causing further overhead on standardizing and implementing these migrations.

text: Furthermore, as with any new proposal in cryptography, there will be some uncertainty with the actual security levels offered by proposed PQCsystems, at least in the early stages. A scenario where a new cryptanalytical attack is discovered after we have migrated away to PQC, from current cryptography, such as modern Elliptic Curve Cryptography (ECC) systems, is not inconceivable. In an even worse scenario, such an attack might not require quantum computations at all, making our communications and infrastructure weaker here and now.

text: Therefore early adopters will need to make a trade-off between quantum-resistance (i.e. migrating to a PQC system), current security-levels attested by years of research (i.e. keep using a current proven cryptosystem, such as ECC) and financial, time or computational resources (i.e. deploy and use a double encryption or signature scheme), at least till PQC systems mature and the advent of new security models, new proof techniques, and supporting tools allow us to identify [cryptographic] constructions that cannot be proven secure and replace them.

text: This study provides an overview of the state-of-the-art with regard to the challenges awaiting us, beyond the design and selection of a PQC system.

text: In the world of post-quantum cryptography, protocol designers need to be aware of the possibility of different trade-offs andchoose systems matching their application scenario, taking into account (1) how frequently public keys are sent, relative to ciphertexts or signed messages using them, and (2) how important computation speed is relative to bandwidth. Developing guidelines for major use cases will ease the process.

text: Existing protocols might need tweaks and updates to ensure proper integration with PQC. It makes sense to withhold major changes till we decide on one or more PQC systems

text: 27 POST QUANTUM CRYPTOGRAPHY Integration study October 2022

text: A different approach to dealing with the difference in interfaces is to design new protocols, developed around post-quantum systems and taking their specifications into account. This option will probably lead to the most efficient protocols, but with a greater cost. Examples covered are: replacing signatures with KEMs, putting key-sizes as guiding factors, and offering post-quantum protection at a different layer of the Internet. The latter has the added benefit of promoting a fast rollout, not hampered by slow standardisation processes. The number of published proposals for new protocols built specifically around PQC systems is still somewhat limited and more work in this direction is required.

text: We already mentioned the uncertainty of migrating to a new cryptosystem. To address this risk one can use a hybrid system; i.e. deploy post-quantum cryptography today as an extra layer together with pre-quantum cryptography, rather than deploying it as a replacement for pre-quantum cryptography. Sign with a pre-quantum cryptosystem and with a post-quantum cryptosystem, making sure that the verifier checks both signatures. Encrypt as usual with pre-quantum cryptography, and then encrypt the result with postquantum cryptography, so that decryption requires both secret keys.

text: Applications that can afford the data sent by post-quantum cryptography are likely to be able to afford an extra 32 bytes for double encryption with elliptic-curve cryptography and post-quantum cryptography. Similar comments apply to double signing.

text: However, hybrid systems can be a greater performance problem in other cost metrics. There has been some investigation of compact implementations of post-quantum cryptography for small devices, and there are various ideas on how to save space by merging implementations of post-quantum cryptography with implementations of elliptic-curve cryptography. This is another area where more work is required.

text: At some point the security analysis of post-quantum cryptosystems will eventually stabilize and provided quantum cryptanalysis shows some practical results (i.e. computing elliptic-curve discrete logarithms of considerable length) we will need to revisit the continued value of deploying elliptic-curve cryptography. Assuming that there is public consensus someday that elliptic-curve cryptography is obsolete, it will then be reasonable to consider removing it from protocols such as TLS, with the goal of eventually allowing simpler TLS implementations, and from any hybrid implementations.

text: Based on existing knowledge of quantum attacks, the consensus is that only public key cryptography is threatened, while attacks against other schemes and protocols are not known for now (with symmetric systems a duplication of the key size would defeat current best quantum cryptanalysis). A common approach in modern cryptography is to reduce the attack surface against the mathematical security of a protocol or system to breaking the security of its basic building blocks via security proofs. Building blocks in this case can either be mathematical problems, such as the RSA problem, or higher level building blocks; such as key encapsulation mechanisms (KEM), signature schemes, or hash functions for which we have already established security by another proof or cryptanalysis.

text: Clearly, if new protocols are constructed these require new proofs. But even existing proofs for well-known protocols have to be revisited. The reason is that when we aim for security against adversaries making use of a quantum computer, we have to model our adversaries also as quantum algorithms. This requires changing models and deciding about the specific abilities of quantum adversaries. Existing proofs will have to be vetted again in this new setting. Beyond basic primitives, new models and proofs are still sparse; especially in the analysis of protocols. More research on new models and proof

text: 28 POST QUANTUM CRYPTOGRAPHY Integration study October 2022

text: techniques for these models is required. This will enable us to justify parameter choices for post-quantum cryptography by security proofs.

text: Finally, it should be noted that standardisation bodies continue to standardise protocols built using pre-quantum systems that will not withstand quantum attacks. It is advisable to apply the concepts of hybrid systems (double encryption, double KEM, double signatures, etc.) to such advanced protocols. However, this is not on the current roadmap. Worse, the existence of a recent standard can make it a lot harder to motivate an organization or the community to develop a new standard. Therefore, it is recommend that post-quantum integration should be considered whenever developing new standards or updating existing ones.

text: 29 POST QUANTUM CRYPTOGRAPHY Integration study October 2022

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text: ABOUT ENIS A The European Union Agency for Cybersecurity, ENISA, is the Union’s agency dedicated to achieving a high common level of cybersecurity across Europe. Established in 2004 and strengthened by the EU Cybersecurity Act, the European Union Agency for Cybersecurity contributes to EU cyber policy, enhances the trustworthiness of ICT products, services and processes with cybersecurity certification schemes, cooperates with Member States and EU bodies, and helps Europe prepare for the cyber challenges of tomorrow. Through knowledge sharing, capacity building and awareness raising, the Agency works together with its key stakeholders to strengthen trust in the connected economy, to boost resilience of the Union’s infrastructure, and, ultimately, to keep Europe’s society and citizens digitally secure. More information about ENISA and its work can be found here: www.enisa.europa.eu. ISBN 978-92-9204-590-6 doi: 10.2824/151162