# Basic Research Project J1-1694: Designing certain perfect discrete combinatorial objects in spectral domain

Program Title: |
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Program PI: | Enes Pasalic | ||

Program Code: | J1-1694 | ||

Funding Organization: | Slovenian Research Agency (ARRS) | ||

Research Field (ARRS): | 1.01.00 - Natural Sciences and Mathematics / Mathematics | ||

Duration: | 1.7.2019 - 30.06.2022 | ||

Project Category: | B | ||

Yearly Range: | 0.83 FTE. | ||

Sicris profile of the program: | available here. | ||

| The main goal of this project is to investigate the existence, design and classification of certain discrete combinatorial objects that correspond to a special class of polynomials over finite fields which play an important role in cryptography. Namely, a nonlinear mapping of an n-bit block to an m-bit block (so-called a substitution box aka S-box) is an essential cryptographic primitive whose most prominent usage regards the design of a family of symmetric encryption algorithms, commonly termed block ciphers. In particular, two classes of these nonlinear mappings are of immense importance with respect to the two well established cryptanalytic methods differential and linear cryptanalysis. These classes of functions are respectively called APN (almost perfect nonlinear) and AB (almost bent) functions. The former class consists of mappings from $GF(2)^n$ to $GF(2)^n$ that are characterized with the property that their derivatives, that is an equation $F(x+a)+F(x)=b$ over $GF(2)^n$, admit either 0 or 2 solutions for any nonzero $a$ and any $b$ in $GF(2)^n$. A simple requirement that an APN mapping $F$ is also a permutation, for even $n$, leads to an extremely difficult problem (commonly called BIG APN problem which has been open for last 40 years) of finding such mappings for even n ) 6. The latter class of AB functions over $GF(2)^n$, which exist only for odd $n$ and are necessarily APN, achieve the largest possible resistance to linear cryptanalysis. Apart from their applications in cryptography, these functions are also very closely related to design theory and relative difference sets as pointed out by A. Pott in his survey [18]. |