E-th roots and static Diffie-Hellman using index calculus
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1 E-th roots and static Diffie-Hellman using index calculus Antoine Joux 1 Joint work with Reynald Lercier 2, David Naccache 3, Emmanuel Thomé 4 Elliptic Curve Cryptography 2008 Utrecht 1 DGA and UVSQ 2 DGA and IRMAR 3 ENS 4 INRIA Lorraine 1 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
2 Key questions Security of plain RSA Diffie-Hellman? Factoring Discrete Log. 2 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
3 Quick reminder: RSA RSA: Rivest, Shamir, Adleman (1977) Public key: N a large integer, e encryption exponent Private key: N = pq, p and q prime, d decryption exponent ed = λ(p 1)(q 1) + 1. Encryption : x x e (mod N) Decryption : y e y (mod N) y y d (mod N) 3 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
4 Quick reminder: Diffie-Hellman Invented by Diffie and Hellman (1976) Public parameters: p a large prime, g a generator (subgroup) Key exchange: Alice Bob Choose a g a g b Choose b g ab When a = s is fixed: Static Diffie-Hellman 4 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
5 Quick reminder: RSA and factoring? Pros: Finding d is as difficult as factoring N Probabilistic (already in RSA from Miller 1975) Deterministic (May 2004) Breaking RSA may be as difficult as factoring (Brown 2006) Cons: Specific weaknesses: Multiplicative attacks Blinding Breaking RSA may be easier than factoring (Boneh, Venkatesan, 1998) 5 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
6 Specific weaknesses Multiplicative attacks: From e a and e b, deduce e ab. Blinding: Ask e ar e. Deduce e a. 6 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
7 Quick reminder: Diffie-Hellman and DLOG? Computational Diffie-Hellman and Discrete Log. (Maurer-Wolf 1996) Static Diffie-Hellman less clear (Brown-Gallant 2005) 7 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
8 Reformulating the key question RSA: Given access to an e-th root oracle: Can we learn to compute e-th roots? Efficienty (with a cost lower than factoring)? Diffie-Hellman Given access to a static Diffie-Hellman oracle: Can we learn to raise to the secret power? Efficienty (with a cost lower than discrete log.)? 8 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
9 Reminder: Number Field Sieve Z[X] Q(α 1 ) Q(α 2 ) Z/NZ or Z/pZ Number fields defined from two polynomials: f 1 and f 2 Relies on multiplicative relations over smoothness bases Applicable to factoring and discrete logarithms Complexity: L N (1/3, (64/9) 1/3 ) = e ((64/9)1/3 +o(1)) log 1/3 N log log 2/3 N) 9 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
10 Reminder: simplified Function Field Sieve Z/pZ[X, Y ] Z/pZ[X] Z/pZ[Y ] F p n Function fields defined from two polynomials: x = f 1 (y) and y = f 2 (x) Applicable to discrete logarithms in small characteristic Complexity: L N (1/3, (32/9) 1/3 ) = e ((32/9)1/3 +o(1)) log 1/3 N log log 2/3 N) 10 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
11 Reminder 1 : NFS and FFS 1. Find smooth objects and write multiplicative relations 2. Do linear algebra 3. Final stage Finish factorization: Square root of ideal (Montgomery) Compute individual discrete logarithms: Descent 1 Another reminder: both are heuristic algorithms 11 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
12 A special case for RSA: Affine modular roots (AMR) Special oracle e c + x (c fixed, x small) Multiplicative attack? Known attacks when x N 1/3 is allowed Arbitrary e-th roots? Q(α 1 ) Z[X] e Z/NZ 12 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
13 A special case for RSA: Affine modular roots 1. One sided smooth objects: multiplicative relations with e 2. Do linear algebra: e of basis elements 3. Final stage Get multiplicative relation Existential forgery Compute arbitrary e-th roots (with additional queries) Universal forgery One sided descent 13 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
14 Answering the key question General oracle e x or x s Collect two sides Sieving on one side. Twice. Same complexity! Z[X] Z[X] Q(α 1 ) e e Q(α 2 ) Z/NZ Z/NZ 14 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
15 Easy case: FFS in small characteristic Two linear sides: No sieving and no linear algebra Descent (compute s-th power for h(x)) Randomize until: h(x) = A(x) B(x) is smooth enough. For each factor q(x), choose l(x, y) to find: q(x)c(x) = D(y), with C(x) and D(y) smooth enough Finally backtrack from known s-th powers 15 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
16 Special q: How to We want q(x) to divide l(x, f 2 (x)) That s deg(q) linear conditions Precompute 1, x,..., x dx (modulo q(x)) Precompute f 2 (x), xf 2 (x),..., x dx f 2 (x). Precompute f 2 (x) dy, xf 2 (x) dy,..., x dx f 2 (x) dy Construct matrix and find kernel 16 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
17 FFS experiment in F Two polynomials: y = x x 4 + x 3 + x x = y 6 + y millions calls to oracle (deg. up to 29) Total runtime less than a week (single computer 2 ) For details, see IACR eprint Intel Core-2 at 3.6 GHz 17 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
18 General case 1. Collect relations: On side 1: sieving On side 2: directly obtain e-th roots or s-th powers 2. Do linear algebra: e or s-th powers of basis elements (side 1) Possibly delayed 3. Optionally enlarge smoothness bases 4. Final stage Descent as in discrete logs Recover e-th root or s power 18 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
19 General case: linear algebra Type of linear algebra: Modulo e (or p 1) with Schirokauer s maps Alternatively: Exact Before the final stage: Multiplicative Or postponed to backtrack of final phase 19 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
20 General case: Descent Descent for H: Randomize until: H = A B is smooth enough in Z. For each factor q, choose ax + b to find: q.c = D(α), with C and norm of D(α) smooth enough Backtrack (postponed linear algebra here) If modulo e, need variant of Montgomery s square root 20 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
21 RSA experiment on 512 bits With public exponent e = millions calls to oracle Initial sieving: 2 CPU hours 1 Bases extension: 44 CPU hours Descent time: around one hour Linear algebra 2 : 6 hours on 4 proc. Montgomery e-th root: five minutes For details, see IACR eprint Reminder: Factoring this number took 8000 mips.years 1 AMD Opteron 2.4GHz. 2 Intel Core 2 at 2.667GHZ 21 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
22 Dlog experiment on 516 bits Using p = π millions calls to oracle Initial sieving: 4 minutes on 128 proc. 1 (FB 2 19 ) Base extension: 24 more minutes (FB 2 32 ) Linear algebra 2 : 8 hours on 4 proc. Descent time: around two hours For details, see IACR eprint Intel Core 2 at 1.6 GHz 2 Intel Core 2 at 2.4 GHZ 22 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
23 Asymptotic complexity All complexities are L(1/3, 3 ): variant calls lin. alg. descent Dlog FFS 4/9-4/9 32/9 NFS-HD 48/91 384/91 384/91 128/9 NFS 3 4/9 32/9 3 64/9 Reminder, range of algorithms: Algorithm From p To p FFS 2 L p n(1/3, ) NFS-HD L p n(1/3, ) L p n(2/3, ) NFS L p n(2/3, ) p = p n 3 Requires Montgomery algorithm for RSA 23 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
24 Open problems Use oracle to factor/compute discrete log. faster? What about static Diffie-Hellman on curves? 24 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
25 Conclusion Questions? 25 A. Joux, R. Lercier, D. Naccache, E. Thomé E-th roots and static Diffie-Hellman using index calculus
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