A New Class of Q-Ary Codes for the McEliece Cryptosystem

Abstract

The McEliece cryptosystem is a promising candidate for post-quantum public-key encryption. In this work, we propose q-ary codes over Gaussian integers for the McEliece system and a new channel model. With this one Mannheim error channel, errors are limited to weight one. We investigate the channel capacity of this channel and discuss its relation to the McEliece system. The proposed codes are based on a simple product code construction and have a low complexity decoding algorithm. For the one Mannheim error channel, these codes achieve a higher error correction capability than maximum distance separable codes with bounded minimum distance decoding. This improves the work factor regarding decoding attacks based on information-set decoding.

Generalized Concatenated Codes over Gaussian and Eisenstein Integers for Code-Based Cryptography

“…In [23] a product code construction from outer Reed-Solomon (RS) and inner one, error correcting (OEC) codes over Gaussian integers was proposed. In the following, we review this code construction.…”

“…We consider an outer RS code For instance, consider the special case of the inner OEC codes of length n i = 2 and minimum Mannheim distance d i = 4 [23]. These codes are generated by a field element a of weight at least three.…”

“…In [23], product codes of outer RS codes and inner one-Mannheim error correcting (OMEC) codes were proposed for the McEliece system. Those codes are defined over Gaussian integers, which are complex numbers with integers as real and imaginary parts [24,25].…”

“…A higher error correction capability leads to a higher work factor for comparable parameters. On the other hand, the concatenated codes presented in [23] can be attacked with a combination of the structural attacks from [21,22].…”

“…We investigate the weight-one error channel for Eisenstein integers in Section 4. In Section 5, we adapt the product codes from [23] to Eisenstein integers. The new code construction based on generalized concatenated codes is discussed in Section 6.…”