By Chuan-Kun Wu, Dengguo Feng
This ebook specializes in different representations and cryptographic homes of Booleans capabilities, offers buildings of Boolean services with a few reliable cryptographic houses. extra particularly, Walsh spectrum description of the normal cryptographic homes of Boolean capabilities, together with linear constitution, propagation criterion, nonlinearity, and correlation immunity are offered. buildings of symmetric Boolean capabilities and of Boolean variations with strong cryptographic homes are particularly studied. This e-book isn't really intended to be entire, yet with its personal concentrate on a few unique examine of the authors long ago. To be self content material, a few simple recommendations and homes are brought. This booklet can function a reference for cryptographic set of rules designers, quite the designers of flow ciphers and of block ciphers, and for teachers with curiosity within the cryptographic houses of Boolean functions.
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Extra info for Boolean Functions and Their Applications in Cryptography
Let S be a set of all the coset leaders of V ? ˛/ D ˛H where H is the matrix as described in the theorem. ˛1 ˛2 /H D 0. ˛1 ˛2 / 2 V ? ; therefore, ˛1 D ˛2 ; hence, ' is a one-to-one mapping. 2/, hence ' must be a bijection. 2/, there must exist ˛ 2 S and ˇ 2 V ? x/ has an algebraically degenerated function in Fk . 2. xA/. f /. 10. x/ 2 Fn . x/ can be algebraically degenerated into a function in k variables. x1 ; x2 ; : : : ; xn / holds. 2/ ! 2/ as: x ! x/. ˛i ˚ x/. ˛i D/. 1/hw; xi . ˛i D/. 1/hw; ˛i i iD1 D 2n 0 X .
Shannon has proved that, in order to achieve one-time-pad security, the size of the key should be at least the same as that of the message to be encrypted. When the message is large, to allow both of the communication parties to share a key of the required size can be difficult. In order to make it more practical, one solution is to use pseudorandom sequences instead of pure random sequences as encryption keys. A pseudorandom sequence is defined to have the following properties: (1) randomness and (2) reproduction.
However, the algebraic degree of the employed Boolean functions cannot be too low either. In general, the algebraic degree should be larger than n2 . , the chances for the output to be 0 are about the same as the chances for the output to be 1. When a Boolean function has equal chances to output 0 or 1 when the input variables go through all the possibilities, the function is called balanced. f / D 2n 1 . In the design of stream ciphers, Boolean functions are often required to be balanced, as this would give outputs with balanced number of 0’s and 1’s and looks more random.