Survey on recent trends towards generalized differential and boomerang uniformities

Differential cryptanalysis is a general form of cryptanalysis applicable primarily to block and stream ciphers and cryptographic hash functions. The discovery of differential cryptanalysis is generally attributed to Biham and Shamir in the late 1980s, who published several attacks against various block ciphers and hash functions, including a theoretical weakness in the Data Encryption Standard (DES). Boomerang cryptanalysis is a method for the cryptanalysis of block ciphers based on differential cryptanalysis. It was invented by Wagner in (FSE, LNCS 1636, 156–170, 1999) and has allowed new avenues of attack for many ciphers previously deemed safe from differential cryptanalysis. Differential and boomerang uniformities are crucial tools to handle and analyze vectorial functions (designated by substitution boxes, or briefly S-boxes in the context of symmetric cryptography) to resist differential and boomerang attacks, respectively. Ellingsen et al. (IEEE Transactions on Information Theory 66(9), 2020) introduced a new variant of differential uniformity, called c-differential uniformity (where c is a non-zero element of a finite field of characteristic p), of p-ary (n, m)-function for any prime p obtained by extending the well-known derivative of vectorial functions into the (multiplicative) c-derivative. Later, Stănică [Discrete Applied Mathematics, 2021] introduced the notion of c-boomerang uniformity. Both c-differential and c-boomerang uniformities have been extended to the idea of simple differential and boomerang uniformities, respectively, which are recovered when c equals 1.This survey paper combines the known results on this new concept of differential and boomerang uniformities and analyzes their possible cryptographic applications. This survey presents an overview of these significant concepts that might have greater implications for future theoretical research on this subject and applied perspectives in symmetric cryptography and related topics. Along with the paper, we analyze these discoveries and the results provided synthetically. The article intends to help readers explore further avenues in this promising and emerging direction of research. At the end of the article, we present more than nine lines of perspectives and research directions to benefit symmetric cryptography and other related domains such as combinatorial theory (namely, graph theory).