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

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

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

Our results Security level: 2-k Trapdoor perm size: kT = poly(k) Circuit size: |C| = poly(k) Witness size: |w|  |C| CRS in bits Proof in bits Assumption Kilian-Petrank |C|∙kT∙k∙(log k) Trapdoor perms This work |C|∙kT∙polylog(k) CRS in bits Proof in bits Assumption Gentry poly(k) |w|∙poly(k) Lattice-based G-Ostrovsky-Sahai k3/polylog(k) |C|∙k3/polylog(k) Pairing-based This work |C|∙polylog(k) Naccache-Stern

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

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

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

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

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)m Chinese remainder gives m mod P

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) 01...0 c1 Epk(101;r2) 10? ; 2 11…1 c2 Epk(011;r3) ??1 ; 3 00…1 c3 K(1k)  (pk,sk) ??? ; 4 Epk(110;r4) 10…0 c4

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

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

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

Probably hidden pairs are 00 and 11 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

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

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

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

Summary Technique 1: Reduce soundness error with probabilistically checkable proofs Technique 2: Implement hidden random bit string with Naccache-Stern encryption Hidden bits Proof in bits Assumption Kilian-Petrank |C|∙kT∙k∙(log k) Trapdoor perms This work |C|∙kT∙polylog(k) CRS in bits Proof in bits Assumption Gentry poly(k) |w|∙poly(k) Lattice-based G-Ostrovsky-Sahai k3/polylog(k) |C|∙k3/polylog(k) Pairing-based This work |C|∙polylog(k) Nacache-Stern