Improved Related-Key Attacks on DESX and DESX+

In this paper, we present improved related-key attacks on the original DESX, and DESX+, a variant of the DESX with its pre- and post-whitening XOR operations replaced with addition modulo $2^{64}$. Compared to previous results, our attack on DESX has reduced text complexity, while our best attack on DESX+ eliminates the memory requirements at the same processing complexity.