Autocorrelations of Vectorial Boolean Functions

Abstract

Recently, Bar-On et al. introduced at Eurocrypt'19 a new tool, called the differential-linear connectivity table (DLCT), which allows for taking into account the dependency between the two subciphers E0 and E1 involved in differential-linear attacks. This paper presents a theoretical characterization of the DLCT, which corresponds to an autocorrelation table (ACT) of a vectorial Boolean function. We further provide some new theoretical results on ACTs of vectorial Boolean functions.

Further Results on Autocorrelation of Vectorial Boolean Functions

“…Let 𝐹 be a vectorial Boolean function, for any 𝑢 ∈ F 𝑛 2 and 𝑣 ∈ F 𝑚 2 , when only 𝑢 is variable, the sum-ofsquares indicator is 𝜎 𝐹 (𝑢) = 𝑣∈F 𝑚 2 𝐴𝐶 2 𝐹 (𝑢, 𝑣). The relationship between the differential distribution table and the autocorrelation of vectorial Boolean functions was presented in [4], which will be used to the proof of Theorem 4. Lemma 2.…”

“…Lemma 2. [4] Let 𝐹 be a vectorial Boolean function. Then, for any 𝑢 ∈ F 𝑛 2 and 𝑣 ∈ F 𝑚 2 , we have ∑︁…”

“…Definition 5. [4] Let 𝑛, 𝑚 be two positive integers. There exist two spectrums of the vectorial Boolean functions 𝐹.…”

“…In [3], Zhang and Zheng introduced the GAC of Boolean functions by sum-of-squares indicator and absolute indicator which are constructed by autocorrelation function. In [4], Canteaut et al introduced the autocorrelation function of vectorial Boolean functions by the autocorrelation table(ACT) [5] and constructed the absolute indicator of vectorial Boolean functions by autocorrelation function. Inspired by [3]- [5], we give three kinds of the sum-of-squares indicators of vectorial Boolean functions, and obtain their upper and lower bounds, respectively.…”

Investigations of c-Differential Uniformity of Permutations with Carlitz Rank 3

“…For even characteristic, some cryptographic properties of permutations of low Carlitz rank have been investigated in several researches, and we can see that they have good cryptographic parameters. For example, the multiplicative inverse function of Carlitz rank 1 has low differential uniformity [17], high nonlinearity [12], low boomerang uniformity [5], low differential-linear uniformity [6], low cdifferential uniformity [7], and low c-boomerang uniformity [18]. Furthermore, it is used as the S-box of the AES(Advanced Encryption Standard) cryptosystem.…”