Differential Factors Revisited: A Sufficient Condition for the Practical Use of Differential Factors

Differential factors, introduced by Tezcan and Özbudak at LightSec 2014, are properties of the S-boxes that equalize the counters of some guessed keys, thereby reducing the key space for the key guess process. Differential factors have been used to reduce the key space for the attacks on SERPENT, PRESENT, PRIDE, and RECTANGLE. In this paper, we demonstrate that some differential factors do not actually reduce the key space for the differential-linear attack on SERPENT and the relatedkey differential attack on RECTANGLE. Moreover, by comparing these instances with the differential attack on PRESENT, where differential factors do have an effect, we identify a sufficient condition for the practical use of differential factors. This condition enables preemptive identification of differential factors that could impact the key space for attacks on other ciphers.