## Cryptology ePrint Archive: Report 2010/585

**Smaller decoding exponents: ball-collision decoding**

*Daniel J. Bernstein and Tanja Lange and Christiane Peters*

**Abstract: **Very few public-key cryptosystems are known that can encrypt and decrypt in time b^(2+o(1)) with conjectured security level 2^b against conventional computers and quantum computers. The oldest of these systems is the classic McEliece code-based cryptosystem.
The best attacks known against this system are generic decoding attacks that treat McEliece’s hidden binary Goppa codes as random linear codes. A standard conjecture is that the best possible w-error-decoding attacks against random linear codes of dimension k and length n take time 2^((alpha(R,W)+o(1))n) if k/n goes to R and w/n goes to W as n goes to infinity.
Before this paper, the best upper bound known on the exponent alpha(R, W) was the exponent of an attack introduced by Stern in 1989. This paper introduces “ball-collision decoding” and shows that it has a smaller exponent for each (R, W): the speedup from Stern’s algorithm to ball-collision decoding is exponential in n.

**Category / Keywords: **McEliece cryptosystem, Niederreiter cryptosystem, post-quantum cryptography, attacks, information-set decoding, collision decoding

**Date: **received 17 Nov 2010, last revised 7 Mar 2011

**Contact author: **c p peters at tue nl

**Available format(s): **PDF | BibTeX Citation

**Version: **20110307:183201 (All versions of this report)

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