Improvements of Algebraic Attacks Based on Structured Gaussian Elimination

Algebraic attacks are studied as a potential cryptanalytic procedure for various types of ciphers. The XL SGE algorithm has been recently proposed to improve the complexity of the XL attack. XL SGE uses structured Gaussian elimination (SGE) during the expansion phase of XL. In this paper, we establish that XL SGE suffers from some serious drawbacks that impair the effectiveness of SGE-based reduction at all multiplication stages except the first. In order to avoid this problem, we propose several improvements of XL SGE. Our modifications are based upon partial monomial multiplication and handling of columns of weight two. Our modified algorithms have been experimentally verified to be substantially superior to XL SGE.