Speeding up deciphering by hypergraph ordering

Abstract

The Gluing Algorithm of Semaev (Des. Codes Cryptogr. 49:47---60, 2008)--that finds all solutions of a sparse system of linear equations over the Galois field $$GF(q)$$ G F ( q ) --has average running time $$O(mq^{\max \left| \cup _{1}^{k}X_{j}\right| -k}),$$ O ( m q max ? 1 k X j - k ) , where $$m$$ m is the total number of equations, and $$\cup _{1}^{k}X_{j}$$ ? 1 k X j is the set of all unknowns actively occurring in the first $$k$$ k equations. In order to make the implementation of the algorithm faster, our goal here is to minimize the exponent of $$q$$ q in the case where every equation contains at most three unknowns. The main result states that if the total number $$\left| \cup _{1}^{m}X_{j}\right| $$ ? 1 m X j of unknowns is equal to $$m$$ m , then the best achievable exponent is between $$c_{1}m$$ c 1 m and $$c_{2}m$$ c 2 m for some positive constants $$c_{1}$$ c 1 and $$c_{2}.$$ c 2 .