By Antoine Joux

Illustrating the ability of algorithms, Algorithmic Cryptanalysis describes algorithmic equipment with cryptographically correct examples. targeting either inner most- and public-key cryptographic algorithms, it offers each one set of rules both as a textual description, in pseudo-code, or in a C code program.

Divided into 3 elements, the publication starts with a quick creation to cryptography and a history bankruptcy on effortless quantity concept and algebra. It then strikes directly to algorithms, with each one bankruptcy during this part devoted to a unmarried subject and infrequently illustrated with easy cryptographic purposes. the ultimate half addresses extra refined cryptographic purposes, together with LFSR-based move ciphers and index calculus methods.

Accounting for the impression of present machine architectures, this booklet explores the algorithmic and implementation elements of cryptanalysis tools. it could function a guide of algorithmic tools for cryptographers in addition to a textbook for undergraduate and graduate classes on cryptanalysis and cryptography.

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Example text

In that case, we need to write p − 1 = 2e q with q odd. The method of choice to compute square roots in this case is Shanks-Tonelli algorithm. This method is based on the remark that for any quadratic residue z = u2 (mod p), the value z (q+1)/2 is “almost” a square e root for z. More precisely, if we let θ = z (q+1)/2 /u, then θ2 = 1. Indeed: θ 2e = 2e uq+1 u e = u2 q = up−1 = 1 (mod p). 18) As a consequence, to obtain a correct square root, it suffices to multiply z (q+1)/2 by an adequate 2e -th root of unity in Fp .

Thus, somewhere in the path between y and 1, we encounter a square root of 1. If this square root is trivial or in the rare case where y = 1, we simply choose another value for x. In truth, the knowledge of φ(N ) is not really needed to use this argument: it is also possible to factor N using the same method when a multiple of φ(N ) is known. Also note that in the two-factor case, there is an easy deterministic method to factor N when φ(N ) is known. Indeed, in that case, we know that pq = N and p + q = N + 1 − φ(N ).

Remember that in that case, GCD(A, 2B) = GCD(A, B). If β is even, the coefficients are simply (α , β /2). If β is odd, they are (α + B, (β + A)/2). Note that there exist algorithms for computing GCDs with a better asymptotic complexity than Euclid’s or Stein’s algorithm. 1 yields more information about the relation between its inputs X and Y than simply its outputs GCD(X, Y ), (αy , βy ) and (αx , βx ). e. 1. More precisely, we have: z0 βi zi 1 + = ≤ . 9) z1 αi αi z1 |αi αi+1 | As a consequence, it can be very useful to use intermediate values of αi , βi and zi to obtain a small linear combination of z0 and z1 whose coefficients are also small.

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