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Differential Factors Revisited: Corrected Attacks on PRESENT and SERPENT

Differential factors, which prevent the attacker to distinguish some of the guessed keys corresponding to an active S-box during a differential attack on a block cipher, are recently introduced at Lightsec 2014 and used to reduce the time complexities of the previous differential-linear attacks on Serpent. Key recovery attacks generally consists of two parts: Key guess using the distinguisher and exhaustive search on the remaining key bits. Thus, we show that differential factors can reduce the time complexity of the former and increase the latter since the attacker does not need to guess the keys which cannot be distinguished. As an example for the latter, we show that the best known differential attack on Present overlooked its six differential factors and the corrected attack actually requires a time complexity increased by a factor of 64. Moreover, we show that differential factors also reduce data complexity of the differential attacks since less number of pairs are required to distinguish the correct key when the key space is reduced. This reduction in data complexity also reduces the time complexity. By using Serpent's differential factors, we further reduce the data and time complexity of the differential-linear attacks on this cipher to obtain the best attacks.