On the Differential Linear Connectivity Table of Vectorial Boolean Functions

Vectorial Boolean functions are crucial building blocks in symmetric ciphers. Different known attacks on block ciphers have resulted in diverse cryptographic criteria of vectorial Boolean functions,such as differential distribution table and nonlinearity. Very recently, Bar-On et al. introduced at Eurocrypt'19 a new tool, called the Differential-Linear Connectivity Table (DLCT).This paper is a follow-up work, which presents further theoretical characterization of the DLCT of vectorial Boolean functions and also investigates this new criterion of functions with certain forms. In this paper we introduce a generalized concept of the additive autocorrelation, which is extended from Boolean functions to the vectorial Boolean functions, and use it as a main tool to investigate the DLCT property of vectorial Boolean functions. Firstly, by establishing a connection between the DLCT and the additive autocorrelation, we characterize properties of DLCT by means of the Walsh transform and the differential distribution table, and present generic lower bounds on the differential-linear uniformity (DLU) of vectorial Boolean functions. Furthermore, we investigate the DLCT property of monomials, APN, plateaued and AB functions. Our study reveals that the DLCT of these special functions are closely related to other cryptographic criteria. Next, we prove that the DLU of vectorial Boolean functions is invariant underthe EA equivalence but not invariant under the CCZ equivalence, and that the DLCT spectrum is only invariant under affine equivalence. In addition, under affine equivalence, we exhaust the DLCT spectra and DLU of optimal S-boxes with $4$ bit by Magma. Finally, we investigate the DLCT spectra and DLU of some polynomials over $F_{2^n}$, including the inverse, Gold, Bracken-Leander power functions and all quadratic polynomials.