Cryptographic properties and application of a Generalized Unbalanced Feistel Network structure

Abstract

In this paper, we study GF-NLFSR, a Generalized Unbalanced Feistel Network (GUFN) which can be considered as an extension of the outer function FO of the KASUMI block cipher. We show that the differential and linear probabilities of any n + 1 rounds of an n-cell GF-NLFSR are both bounded by p ², where the corresponding probability of the round function is p. Besides analyzing security against differential and linear cryptanalysis, we provide a frequency distribution for upper bounds on the true differential and linear hull probabilities. From the frequency distribution, we deduce that the proportion of input-output differences/mask values with probability bounded by p ⁿ is close to 1 whereas only a negligible proportion has probability bounded by p ². We also recall an n ²-round integral attack distinguisher and (n ² + n − 2)-round impossible differential distinguisher on the n-cell GF-NLFSR by Li et al. and Wu et al. As an application, we design a new 30-round block cipher Four-Cell⁺ based on a 4-cell GF-NLFSR. We prove the security of Four-Cell⁺ against differential, linear, and boomerang attack. Four-Cell⁺ also resists existing key recovery attacks based on the 16-round integral attack distinguisher and 18-round impossible differential distinguisher. Furthermore, Four-Cell⁺ can be shown to be secure against other attacks such as higher order differential attack, cube attack, interpolation attack, XSL attack and slide attack.