Exploring FAEST: Post-Quantum Signatures and Zero-Knowledge Proofs
Delve into the world of FAEST, a post-quantum signature scheme, with a focus on publicly verifiable zero-knowledge proofs. The presentation covers VOLE-in-the-Head, families of ZK proofs, and the application of VOLE in creating VOLE-ZK proofs. Learn about the background of VOLE, its use in the designated verifier setting, and how it enables ZK proofs through linear commitment and commitment-prove methods.
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FAEST Emmanuela Orsini Carsten Baum Christian Majenz Lennart Braun Lawrence Roy (Lance) Peter Scholl Sebastian Ramacher Cyprien Delpech de Saint Guilhem Shibam Mukherjee Christian Rechberger Michael Kloo 1
Banquet (post-quantum) Signature dishes Picnic BBQ 2
Based on Publicly Verifiable Zero-Knowledge and Post-Quantum Signatures From VOLE-in-the-Head Carsten Baum, Lennart Braun, Cyprien Delpech de Saint Guilhem, Michael Kloo , Emmanuela Orsini, Lawrence Roy, Peter Scholl CRYPTO 2023 No VOLEs were harmed while making this presentation 4
Families of ZK Proofs Linear MPC-in-the-head VOLE-ZK Proof size Ligero Size: 1 bit per AND gate designated verifier STARKs ... Prover runtime 5
VOLE-in-the-Head: a general tool for making VOLE-ZK proofs publicly verifiable Speed Security Linear size AES/SHA 11-16x witness Application: FAEST post-quantum signature scheme 6
VOLE VOLE- -ZK in the designated verifier setting ZK 7
Background: VOLE (vector oblivious linear evaluation) ?(?) = ?? + ? Prover Verifier ? ? ?? ? VOLE ? ?? ? = ? + ? Can be instantiated with OT, HE, LPN 8
ZK from VOLE (designated verifier) [BMRS 21, WYKW 21] Use VOLE as a linear commitment to ? To open Prover sends (?,?), Verifier checks if? = ? + ? Hiding: since ? is random Binding: opening to ? ? requires guessing prob. 1/|?| (as long as is kept secret from Prover D.V.) ?(?) = ?? + ? ? ? (?) Commitments are linearly homomorphic 9
ZK from VOLE via Commit-and-Prove [BMRS 21, WYKW 21] Commit to witness ? Evaluate ? gate-by-gate: Linear gates: easy Multiplication: ??? 10
Multiplication check (QuickSilver) [DIO 21, YSWW 21] ? ? ? Multiply two lines quadratic polynomial ???? = ?0+ ?1? + ???2 Commit to output ? ??? = ? + ?? ???? ???(?) should be degree-1 Open and check First, mask with random deg-1 commitment ??(?) ?1 ?3= ???( ) ???(?) ?2 ??(?) 11
Cost analysis for VOLE-ZK Per multiplication gate: Commit to ? o 1 VOLE element Open masked commitment o Can be amortized to constant size (check random combination of polynomials) For circuit: ? + ? field elements for ? inputs and ? mult. gates (assuming LPN-based VOLE) Improvements: o General deg-2 and higher degree gates; branching; field switching 12
VOLE VOLE- -in in- -the Adding public verifiability the- -Head Head 13
MPC-in-the-Head vs VOLE-in-the-head: high-level differences ?1 challenge MPC protocol for ?(?1+ + ??) ? 1 views from ? ?? ? = ?1+ + ?? 14
How to do VOLE? Warm-up: using OT Key observation: ?-out-of-? secret sharing + OT VOLE in ?? (from SoftSpokenOT [Roy 22]) ?1 Pick ?? (? 1)-out- of-? OT ? ?? for ? ?? 0 when ? = ? = ?? ( ?) ? = ?1+ + ?? ? = 1 ?1 ? ?? (in ??) ? = ? + ? 15
How to do VOLE-in-the-head? All-but-one vector commitment Commit to ? random strings Run VOLE-ZK proof Challenge Open ? 1 GGM tree (similarly to hypercube) Convert to VOLE ?, ? ? = ? + ? 16
VOLE-in-the-head: some details (? 1)-out-of-? vector commit VOLE in ?? Commitments have soundness error 1 What about ?? for large ?? ? For extension fields, ? = ??: Repeat ? times, with same ? ?? Cost over ?2, 11-16 bits per AND (for 128-bit security) Needs consistency check 17
More details: consistency checking When repeating ? times: Need to ensure prover uses consistent ? Consistency check from[Roy 22]: Universal hash function ? ? = ??, ? = ? ? Check ? ? = ? + ? Security: Needs new analysis for round-by-round soundness with Fiat-Shamir 18
FAEST FAEST Post-quantum signature 19
Paradigm for ZK-based signatures Signature: NIZK proof of knowledge of s?, such that ?? = Enc??? Challenge: finding a ZK-friendly Enc Custom ciphers: e.g. LowMC, MiMC Other assumptions: code-based, multivariate 20
AES: a ZK-friendly OWF? ShiftRows, MixColumns, AddRoundKey: All linear over ?2 S-Box: Inversion in ?28 Prove in ZK as 1 multiplication constraint 21
Proving AES-128 in FAEST Witness: key + internal state of each round (+ key schedule) 1600 bits (in ?2) 200 constraints over ?28: 1 per S-box: degree-2 polynomial: ?? = 1 0 ? 1 (in ?28) if ? = 0 o.w. ? = ? S-Box What if ? = 0? Sample ? such that this never happens 1-2 bits less security (for AES-128) 22
FAEST summary: proving ?? = AES??(?) chall1 chall2 Commit to ?? and ????? after each round ? = ? + ? VOLE consistency check chall3 S-Box Mult. check Open VOLE outputs 23
FAEST performance (AVX2 + AES-NI) Sign/Verify Size 8ms FAEST-128s 5 006 B 1ms FAEST-128f 6 336 B 27ms FAEST-256s 22 100 B 3ms FAEST-256f 28 400 B Timings are from my laptop: AMD Ryzen 7 5800H, 3.2 4.4 GHz Signature sizes: Smaller than SPHINCS+ and most code-based candidates Faster signing, slower verification Even-Mansour variant: Enc?? = AES?? ? 10-15% smaller because of fixed key schedule 24
Work in progress Better domain separation Tighter proofs Parameter tweaking: Smaller signatures Faster verification Other OWFs: Pure MQ: size 3 kB Rain cipher 25
Conclusion Thank you! VOLE-ZK proofs: Lightweight and fast with linear size VOLE-in-the-head: publicly verifiable FAEST signature: Conservative security Reasonable performance 26
