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Short Non-interactive Zero-Knowledge Proofs

Short Non-interactive Zero-Knowledge Proofs. Jens Groth University College London. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A A A A A. Non-interactive zero-knowledge proof . CRS: 0100…11010 Statement: x L. ( x,w )  R L.

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Short Non-interactive Zero-Knowledge Proofs

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  1. Short Non-interactive Zero-Knowledge Proofs Jens Groth University College London TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAA

  2. Non-interactive zero-knowledge proof CRS: 0100…11010 Statement: xL (x,w)RL Proof:  Zero-knowledge: Nothing but truth revealed Soundness: Statement is true Prover Verifier

  3. Non-interactive zero-knowledge proofs Adaptive soundness:Adversary sees CRS before attempting to cheat with false (C,) • Statement C is satisfiable circuit • Perfect completeness • Statistical soundness • Computational zero-knowledge • Uniformly random common reference string • Efficient prover – probabilistic polynomial time • Deterministic polynomial time verifier

  4. Our results • Security level: 2-k • Trapdoor perm size: kT = poly(k) • Circuit size: |C| = poly(k) • Witness size: |w|  |C|

  5. Hidden random string - soundness Statement: xL (x,w)RL 0 1 0 1

  6. Hidden random string – zero-knowledge Statement: xL 0 1

  7. Two new techniques • More efficient use of hidden random bits • Kilian-Petrank: |C|∙k∙(log(k)) hidden random bits • This work: |C|∙polylog(k) hidden random bits • More efficient implementation of hidden bits • Trapdoor permutations: kT = poly(k) bits per hidden random bit • Naccache-Stern encryption: O(log k) bits per hidden random bit

  8. Implementing the hidden random bits model Statement: xL (x,w)RL Epk(0;r1) c1 01...0 c1 Epk(1;r2) c2 11…1 1 ; r2 Epk(0;r3) c3 00…1 c3 K(1k)  (pk,sk) Epk(1;r4) c4 10…0 0 ; r4

  9. Naccache-Stern encryption • pk = (M,P,g) sk = (M) • M is an RSA modulus • P = p1p2…pd where p1,…,pd are O(log k) bit primes • P | ord(g) = (M)/4 and |P| = O(|M|) • Epk(m;r) = gmrP mod M • Dsk(c): For each pi compute m mod pi c(M)/pi = (g(M)/pi)mChinese remainder gives m mod P

  10. Naccache-Stern implementation of hidden bits 0 if m mod pi even 1 if m mod pi odd  if m mod pi is -1 Statement: xL (x,w)RL ?1? ; 1 Epk(010;r1) c1 01...0 10? ; 2 Epk(101;r2) c2 11…1 ??1 ; 3 Epk(011;r3) c3 00…1 K(1k)  (pk,sk) ??? ; 4 Epk(110;r4) 10…0 c4

  11. Revealing part of Naccache-Stern plaintext • Ciphertext c = gmrP • How to prove that m = x mod pi? • Prover reveals  such that P = (cg-x)P/pi • Shows (M) = (gm-xrP)(M)/pi = (g(M)/pi)m-x • Can compute the proof as  = (cg-x)(P-1 mod (M)/P)P/pi • Can randomize proof by multiplying with s(M)/P • Generalizes to reveal m mod iSpi with a proof consisting of one group element

  12. Zero-knowledge • Simulator sets up pk = (M,P,g) such that ord(g) = (M)/4P and g = hP mod M • Simulator also sets up the CRS such that it only contains ciphertexts of the form gt mod M • For any m  ZP we can compute r = ht-m mod M such that gt = gm(gt-m) = gmrP mod M • This means the simulator can open each ciphertext to arbitrary hidden bits

  13. Efficient use of the hidden random bits Statement: xL (x,w)RL 0 1 0 1

  14. Kilian-Petrank Probably hidden pairs are 00 and 11 • Random bits not useful; need bits with structure • Use statistical sampling to get “good” blocks 10 11 00 01

  15. Kilian-Petrank continued • Reveal blocks of bits so remaining “good” blocks of bits have a particular structure (statistically) • Reduce C to a 3SAT formula  • Assign remaining “good” blocks to variables in  • For each clause reveal some bits in the blocks assigned to the literals of the clause • An unsatisfied clause has some probability of the revealed bits not satisfying certain criterion • Repeat many times to make the probability of cheating negligible for each clause

  16. Probabilistically checkable proofs • Polynomial time algorithms f, fw: f: C    belongs to gap-3SAT5 fw: w  x if C(w)=1 then (x)=1 •  is a gap-3SAT5 formula • All variables appear in exactly 5 clauses – thrice as positive literal and twice as negative • Either all clauses are simultaneously satisfiable or a constant fraction are unsatisfiable

  17. Strategy • Compute  = f(C) and prove that it is satisfiable • With the most efficient probabilistically checkable proofs (Dinur 07 combined with BenSasson-Sudan 08) we have || = |C| polylog(k) • Seems counterintuitive to make statement larger • However, since  allows for a constant fraction of “errors” less repetition is needed to make the overall soundness error negligible • It is ok if the prover cheats on some clauses as long as cannot cheat on a constant fraction

  18. Summary • Technique 1: Reduce soundness error with probabilistically checkable proofs • Technique 2: Implement hidden random bit string with Naccache-Stern encryption

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