By Natalia Tokareva
Bent capabilities: effects and purposes to Cryptography offers a distinct survey of the gadgets of discrete arithmetic referred to as Boolean bent services. As those maximal, nonlinear Boolean services and their generalizations have many theoretical and functional functions in combinatorics, coding thought, and cryptography, the textual content offers a close survey in their major effects, proposing a scientific evaluation in their generalizations and purposes, and contemplating open difficulties in class and systematization of bent capabilities.
The textual content is acceptable for rookies and complicated researchers, discussing proofs of numerous effects, together with the automorphism workforce of bent capabilities, the decrease certain for the variety of bent capabilities, and more.
- Provides an in depth survey of bent capabilities and their major effects, featuring a scientific evaluate in their generalizations and applications
- Presents a scientific and precise survey of 1000s of leads to the world of hugely nonlinear Boolean services in cryptography
- Appropriate insurance for college kids from complicated experts in cryptography, arithmetic, and creators of ciphers
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Extra info for Bent functions : results and applications to cryptography
According to Chase et al.  (see also Dillon ), the following theorem holds: Theorem 18. Any quadratic bent function in n variables is extended affinely equivalent to the function x1 x2 ⊕ x3 x4 ⊕ · · · ⊕ xn−1 xn . For bent functions of degree more than or equal to 3, the problem of extended affinely classification is still open. Classifications obtained for small n can be found in Chapter 7. Problems of equivalence of Boolean functions are discussed in more details by Cheremushkin .
The resulting Boolean function for the NFSR of Grain has 19 variables and is cheap to implement for hardware. The best linear approximation gives a linear function in 19 variables that contains at least all the linear terms of the function. There are 214 such functions, and they have bias εg = 2−8 . 36 Bent Functions It is interesting that the filter function h in five variables that transforms four bits from the linear feedback shift register and one bit from the NFSR to a single bit was not designed very carefully by the authors of the cipher.
Indeed, given the vector c of values of a Boolean function f in n variables, PAPR(c) = 21n maxn |Wf (x)|2 . Therefore, bent functions play a substantial x∈F2 role in constructing codes for CDMA systems. In 2007, Zhou et al.  considered codes with low peak-to-average power ratio for multicode CDMA. On the basis of the concept of quarter bent functions, a new inequality relating the minimum order of terms of a bent function and the maximum Walsh spectral magnitude was proven, and it facilitates the generalization of some known results.
Bent functions : results and applications to cryptography by Natalia Tokareva