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An Efficient Decoding of Goppa Codes for the McEliece Cryptosystem

Lim S., Lee H.-S., Choi M.

Fundamenta Informaticae

2014

Vol. 133, nr 4

387--397

The McEliece cryptosystem is defined using a Goppa code, and decoding the Goppa code is a crucial step of its decryption. Patterson’s decoding algorithm is the best known algorithm for decoding Goppa codes. Currently, the most efficient implementation of Patterson’s algorithm uses a precomputation. In this paper, we modify Patterson’s decoding algorithm so that one can remove the precomputation part while sustaining the best efficiency. Precomputations yield additional storage requirement to store the precomputed value which increases as the security level increases in McEliece cryptosystem. In the original decoding algorithm of Patterson, computing square root in a quotient field of polynomial ring over a finite field is necessary. In our modification, the computations are involved only in the arithmetics of polynomial ring over a finite field, not in the quotient field. This achieves better efficiency because one can remove polynomial reductions in the computations of quotient field.

A Complete Generalization of Atkin's Square Root Algorithm

Rotaru A. S., Iftene S.

Fundamenta Informaticae

2013

Vol. 125, nr 1

71--94

Atkin's algorithm [2] for computing square roots in Z*p, where p is a prime such that p ≡ 5 mod 8, has been extended by Müller [15] for the case p ≡ 9 mod 16. In this paper we extend Atkin's algorithm to the general case p ≡ 2s + 1 mod 2s+1, for any s ≥2, thus providing a complete solution for the case p ≡ 1 mod 4. Complexity analysis and comparisons with other methods are also provided.