Augmented sets of output differences and new distinguishers for SPN ciphers

We introduce augmented vector spaces of output differences, new generic and black-box distinguishers for Substitution Permutation Network (SPN) ciphers. Our distinguishers are based on a novel method of constructing a vector of size $$ bits from a given vector of size n bits, where $$ and d is a positive integer. We list all such $$ -bit vectors into a set called the corresponding $$ -order augmented set and define its linear span as the corresponding $$ -order augmented vector space . These sets are related to Reed-Muller codes and we prove that the rank of linear span of $$ -order augmented set is $$ using Reed-Muller codes. We then experimentally estimate the number of n-bit vectors required to span augmented vector spaces of output differences. Following these results, we give a generic and efficient algorithm to compute $$ -order augmented vector space (of difference sets) for substitution permutation network ciphers. We apply our algorithm to lightweight ciphers GIFT, PRESENT and SKINNY and provide in-depth comparison of round-reduced ciphers' distinguishers with random sets. Most notably, our new distinguishers for these ciphers cover more rounds than the subspace trails.