List Decoding for Binary Goppa Codes

Abstract

This paper presents a list-decoding algorithm for classical irreducible binary Goppa codes. The algorithm corrects, in polynomial time, approximately n − p n(n − 2t − 2) errors in a length-n classical irreducible degree-t binary Goppa code. Compared to the best previous polynomialtime list-decoding algorithms for the same codes, the new algorithm corrects approximately t 2 /2n extra errors.

Overview of the Mceliece Cryptosystem and its Security

“…B e r n s t e i n in [5] introduced a list-decoding algorithm for irreducible binary Goppa codes. This algorithm is an extension of Patterson's algorithm that allows the receiver to efficiently decode approximately n n(n − 2t − 2) errors (more than t in a typical setting).…”

Code Based Cryptography and Steganography

“…Symmetrically, adding extra errors makes it possible to use shorter keys while keeping a similar security level, but it also requires the receiver to decode the additional errors. Let t be the error capacity of the code, in [4] authors described a "list decoding algorithm" which can correct up to (n − n(n − 4t − 2))/2) t + 1 errors which is an improvement of a first algorithm decribed in [16]. Since we add extra errors, encrypting distinct codewords can lead to the same cryptogram.…”

“…The size of the public matrix will be 460647 bits instead of 520047 bits, i.e. 12% smaller [16]. Using a list decoding algorithm leads to shorter keys at the expense of a moderately increased decryption time.…”

Simplified High-Speed High-Distance List Decoding for Alternant Codes

“…Of course, this criticism does not apply to bounds that treat , k, as variables, such as the bound O(n 2 / 5 ) in [4,Corollary 5.7]. Furthermore, the "rational" algorithms of [54] and [10] allow a better tradeoff between k, , w and can meaningfully take k, ∈ O(1).…”

“…I took a slightly broader perspective, allowing a large gcd for polynomial values on rational inputs, although at the time I did not see any way to use this extra generality; subsequent applications include [54], [10], and [20].…”