Improved Generic Attacks on Unbalanced Feistel Schemes with Expanding Functions

Abstract

“Generic” Unbalanced Feistel Schemes with Expanding Functions are Unbalanced Feistel Schemes with truly random internal round functions from n bits to (k − 1)n bits with k ≥ 3. From a practical point of view, an interesting property of these schemes is that since n < (k − 1)n and n can be small (8 bits for example), it is often possible to store these truly random functions in order to design efficient schemes (example: CRUNCH cf [6]). Attacks on these generic schemes were studied in [7] and [18]. As pointed in [7] and [18], there are surprisingly much more possibilities for these attacks than for generic balanced Feistel schemes or generic unbalanced Feistel schemes with contracting functions. In fact, this large number of attack possibilities makes the analysis difficult. In this paper, we shall methodically analyze again these attacks. We have created a computer program that systematically analyze all the possible attacks and detect the most efficient ones. We have detected a condition on the internal variables that was not clearly analyzed in [18], and we have found many new improved attacks by a systematic study of all the “rectangle attacks” when k ≤ 7, and then we have generalized these improved attacks for all k. Many simulations on our improved attacks have also been done and they confirm our theoretical analysis.