Exponential lower bound on the highest fidelity achievable by quantum error-correcting codes

On a class of memoryless quantum channels which includes the depolarizing channel, the highest fidelity of quantum error-correcting codes of length n and rate R is proven to be lower bounded by 1-exp[-nE(R)+o(n)] for some function E(R). The E(R) is positive below some threshold R', which implies R' is a lower bound on the quantum capacity.Comment: Ver.4. In vers.1--3, I claimed Theorem 1 for general quantum channels. Now I claim this only for a slight generalization of depolarizing channel in this paper because Lemma 2 in vers.1--3 was wrong; the original general statement is proved in quant-ph/0112103. Ver.5. Text sectionalized. Appeared in PRA. The PRA article is typographically slightly crude: The LaTeX symbol star, used as superscripts, was capriciously replaced by the asterisk in several places after my proof readin

Wild McEliece Incognito

The wild McEliece cryptosystem uses wild Goppa codes over finite fields to achieve smaller public key sizes compared to the original McEliece cryptosystem at the same level of security against all attacks known. However, the cryptosystem drops one of the confidence-inspiring shields built into the original McEliece cryptosystem, namely a large pool of Goppa polynomials to choose from. This paper shows how to achieve almost all of the same reduction in key size while preserving this shield. Even if support splitting could be (1) generalized to handle an unknown support set and (2) sped up by a square-root factor, polynomial-searching attacks in the new system will still be at least as hard as information-set decoding. Furthermore, this paper presents a set of concrete cryptanalytic challenges to encourage the cryptographic community to study the security of code-based cryptography. The challenges range through codes over F2, F3, …, F32, and cover two different levels of how much the wildness is hidden

Simplified high-speed high-distance list decoding for alternant codes

Abstract. This paper presents a simplified list-decoding algorithm to correct any number w of errors in any alternant code of any length n with any designed distance t + 1 over any finite field Fq; in particular, in the classical Goppa codes used in the McEliece and Niederreiter public-key cryptosystems. The algorithm is efficient for w close to, and in many cases slightly beyond, the Fq Johnson bound J ′ = n ′ − √ n ′ (n ′ − t − 1) where n ′ = n(q − 1)/q, assuming t + 1 ≤ n ′. In the typical case that qn/t ∈ (lg n) O(1) and that the parent field has (lg n) O(1) bits, the algorithm uses n(lg n) O(1) bit operations for w ≤ J ′ − n/(lg n) O(1) ; O(n 4.5) bit operations for w ≤ J ′ + o((lg n) / lg lg n); and n O(1) bit operations for w ≤ J ′ + O((lg n) / lg lg n).

Recursive towers of function fields over finite fields

The theory of recursive towers of function fields over finite fields was developed by A. Garcia and the author since 1995. We give a survey about the main ideas and results, and we propose some problems for future work