Smaller Decoding Exponents: Ball-Collision Decoding

Very few public-key cryptosystems are known that can encrypt and decrypt in time b2 + o(1) with conjectured security level 2b 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(α(R,W) + o(1))n if k/n → R and w/n → W as n → ∞. Before this paper, the best upper bound known on the exponent α(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.