Speeding up deciphering by hypergraph ordering

The “Gluing Algorithm” of Semaev [Des. Codes Cryptogr. 49 (2008), 47–60] — that finds all solutions of a sparse system of linear equations over the Galois field GF (q) — has average running time O(mq| k 1Xj |), where m is the total number of equations, and ∪k1Xj is the set of all unknowns actively occurring in the first k equations. Our goal here is to minimize the exponent of q in the case where every equation contains at most three unknowns. The main result states that if the total number |∪m1 Xj | of unknowns is equal to m, then the best achievable exponent is between c1m and c2m for some positive constants c1 and c2.