By Chuan-Kun Wu, Dengguo Feng
This e-book specializes in the several representations and cryptographic homes of Booleans capabilities, offers structures of Boolean features with a few stable cryptographic homes. extra in particular, Walsh spectrum description of the conventional cryptographic homes of Boolean services, together with linear constitution, propagation criterion, nonlinearity, and correlation immunity are awarded. buildings of symmetric Boolean features and of Boolean diversifications with stable cryptographic homes are in particular studied. This e-book isn't intended to be finished, yet with its personal specialize in a few unique study of the authors long ago. To be self content material, a few uncomplicated thoughts and homes are brought. This booklet can function a reference for cryptographic set of rules designers, really the designers of move ciphers and of block ciphers, and for teachers with curiosity within the cryptographic homes of Boolean functions.
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Extra resources for Boolean Functions and Their Applications in Cryptography
X/ is the final output of the generator. This is depicted in Fig. 4. Fig. 5 Cryptographic Properties of Boolean Functions 25 LFSR1 LFSR2 f(x) Output LFSRn Fig. 4 Nonlinear combiner generator There are other LFSR-based generators for pseudorandom sequences such as clock-controlled generators. Since they are not necessary for our introduction to cryptographic properties of Boolean functions, we are not going to introduce them. The nonlinear filtering generators and the nonlinear combiners are somehow equivalent: A nonlinear filtering generator can be treated as a nonlinear combiner with all the LFSRs being the same but different initial states, and a nonlinear combiner can be treated as a nonlinear filtering generator based on a larger LFSR (the length of this hypothetic LFSR should be at least the minimum common divisor of the lengths of the LFSRs in the combiner).
As stated earlier, since there is convenient conversion between type I and type II Walsh transforms, any property given in one type can be converted to the other type of Walsh transform. However, sometimes the expression of certain properties in one type of Walsh transform is more compact than in the other type. The following is one such case where it uses the type II Walsh transform. 8 (Parseval). x/. 25) wD0 Proof. w/ D wD0 n 1 2n 1 2X X Œ . x/Chw; xi wD0 xD0 D n 1 2X yD0 n 1 2n 1 2X X n 1 2X xD0 yD0 wD0 .
X/ must hold, and hence we have Rf . f /. i1 ; i2 ; : : : ; ik / we have Rf . x/ is independent of xi1 ; xi2 ; : : : ; xik . This proves the theorem. 3. x/ 2 Fn . Then Rf . x/, where 1; 2; : : : ; k. 3. When k D 1, the onefold selfcorrelation function is simply called the self-correlation function. 2. Rf . 2/, i D 1; 2; : : : ; k. 2/ whose i-th coordinate is 1 and 0 elsewhere. 4. 3. Here we give a slightly different proof of the sufficiency. Assume Eq. 2/. i1 ; i2 ; : : : ; ik /, there must exist ai 2 f0; 1g such that D a1 ei1 ˚ a2 ei2 ˚ ˚ ak eik .
Boolean Functions and Their Applications in Cryptography by Chuan-Kun Wu, Dengguo Feng