Low Probability Differentials and the Cryptanalysis of Full-Round CLEFIA-128

So far, low probability differentials for the key schedule of block ciphers have been used as a straightforward proof of security against related-key differential analysis. To achieve resistance, it is believed that for cipher with k-bit key it suffices the upper bound on the probability to be 2−k. Surprisingly, we show that this reasonable assumption is incorrect, and the probability should be (much) lower than 2−k. Our counter example is a related-key differential analysis of the well established block cipher CLEFIA-128. We show that although the key schedule of CLEFIA-128 prevents differentials with a probability higher than 2−128, the linear part of the key schedule that produces the round keys, and the Feistel structure of the cipher, allow to exploit particularly chosen differentials with a probability as low as 2−128. CLEFIA-128 has 2 such differentials, which translate to 2 pairs of weak keys. The probability of each differential is too low, but the weak keys have a special structure which allows with a divide-and-conquer approach to gain an advantage of 2 over generic analysis. We exploit the advantage and give a membership test for the weak-key class and provide analysis of the hashing modes. The proposed analysis has been tested with computer experiments on small-scale variants of CLEFIA-128. Our results do not threaten the practical use of CLEFIA.