Decoding Random Linear Codes in

Decoding random linear codes is a fundamental problem in complexity theory and lies at the heart of almost all code-based cryptography. The best attacks on the most prominent code-based cryptosystems such as McEliece directly use decoding algorithms for linear codes. The asymptotically best decoding algorithm for random linear codes of length

n

was for a long time Stern’s variant of information-set decoding running in time

$\tilde{\mathcal{O}}\left(2^{0.05563n}\right)$

. Recently, Bernstein, Lange and Peters proposed a new technique called

Ball-collision decoding

which offers a speed-up over Stern’s algorithm by improving the running time to

$\tilde{\mathcal{O}}\left(2^{0.05558n}\right)$

.

In this paper, we present a new algorithm for decoding linear codes that is inspired by a representation technique due to Howgrave-Graham and Joux in the context of subset sum algorithms. Our decoding algorithm offers a rigorous complexity analysis for random linear codes and brings the time complexity down to

$\tilde{\mathcal{O}}\left(2^{0.05363n}\right)$

.