# Bent Functions: Results and Applications to Cryptography by Natalia Tokareva

By Natalia Tokareva

Bent services: effects and functions to Cryptography

offers a different survey of the gadgets of discrete arithmetic often called Boolean bent capabilities. As those maximal, nonlinear Boolean features and their generalizations have many theoretical and useful purposes in combinatorics, coding conception, and cryptography, the textual content presents a close survey in their major effects, proposing a scientific evaluate in their generalizations and purposes, and contemplating open difficulties in type and systematization of bent features.

The textual content is suitable for beginners and complicated researchers, discussing proofs of numerous effects, together with the automorphism team of bent features, the reduce certain for the variety of bent features, and more.

• Provides a close survey of bent features and their major effects, featuring a scientific review in their generalizations and applications
• Presents a scientific and certain survey of countless numbers of ends up in the world of hugely nonlinear Boolean features in cryptography
• Appropriate assurance for college kids from complicated experts in cryptography, arithmetic, and creators of ciphers

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Additional resources for Bent Functions: Results and Applications to Cryptography

Example text

It is easy to see that there are bent functions of all other possible degrees from 2 to n/2 if n 4 (just use the Maiorana-McFarland construction for this; see Theorem 34). For example, the quadratic Boolean function f (x1 , . . , xn ) = x1 x2 ⊕ x3 x4 ⊕ · · · ⊕ xn−1 xn is bent for any even n. Note that in 2004 Hou [171] determined the bound for p-ary bent functions—namely, he proved that if f is a p-ary bent function (p is prime) (p−1)n + 1. In addition, if f is weakly regular, in n variables, then deg(f ) 2 (p−1)n then deg(f ) 2 .

If n is odd, then everything is completely different. First, the exact upper bound for nonlinearity of a Boolean function in n variables is still unknown! This question is as attractive as it is complicated. Some results on it can be found in articles by Maitra and Sarkar [250] and Kavut et al. [194], among others. Bent Functions: An Introduction 21 A Boolean function is called maximal nonlinear if its nonlinearity is as big as possible. Recall that if n is even, such a definition coincides with the definition of a bent function.

In 1993, Matsui [257] showed that DES is not resistant to linear cryptanalysis. For almost 20 years (from 1980 to 1998), DES was a standard of symmetric encryption in the USA. The weakness of the cipher consisted in the “bad” cryptographic properties of its nonlinear components—S-boxes. Mathematically, an S-box is a vectorial Boolean function that maps n input bits to m output bits. In DES, there are eight distinct S-boxes, completely defined in the standard (they can be easily found, for example, on the Internet).