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By Henry Ibstedt
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05 seconds. Furthermore, the attack still applies even when the secret key is very large (more than 10000 bits). The complexity of the attack is O(w7 d log(p)) which is polynomial with respect to all security parameters. In particular, it is quasi-linear in the size of the secret key which is (2d + 2) log(p). This result is rather surprising since the algebraic attack is often more eﬃcient than the legal decryption algorithm. Keywords: Multivariate Cryptography, Algebraic Cryptanalysis, Section Finding Problem (SFP), Gr¨ obner bases, Decomposition of ideals.
Spaenlehauer 10. Use the CRT to retrieve m = m mod Pi : m = (5t17 + 15t16 + 4t15 + 9t14 + 7t13 + 2t12 + 3t11 + 8t10 + 11t9 + 6t17 + 6t8 + 3t16 + 10t7 + 11t15 + 7t6 + t5 + t13 + 14t4 + 10t12 + 3t3 + 3t11 + 12t2 + 8t10 + 11t + 6t9 + 2)x4 y 4 + (13t8 + 2t7 + 2t6 + 10t5 + 5t4 + 2t3 + 15t2 + 3t + 11). 5 Complexity Analysis In this part, we investigate the complexity of the Level 3 Attack. To simplify the notations, we suppose here that the complexity of multiplying two n × n matrices is O(n3 ). We note that C is a parameter chosen by the attacker.
Generically this Gr¨ obner basis contains only one polynomial Q(y, t). 2: Factor Q = Qi (y, t). Let Q0 (y, t) ∈ Fp [y, t] denote an irreducible factor with highest degree with respect to y. 3: Compute a Gr¨ obner basis of the ideal J = F0 , F1 , X, Q0 . 4: To retrieve the plaintext (up to multiplication by a scalar in Fp ), solve the linear system over Fp (m) dij mijk N FJ (xi y j tk ) = 0. (i,j)∈Λm k=0 If the system has no solution, go back to Step 2 and pick another factor of Q. and deg(J) equations (deg(J) = dim(Fp [x, y, t]/J) when Fp [x, y, t]/J is seen as a Fp -vector space).