Number Theory

Stream Ciphers and Number Theory by Thomas W. Cusick

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By Thomas W. Cusick

This can be the original publication on cross-fertilisations among movement ciphers and quantity idea. It systematically and comprehensively covers recognized connections among the 2 components which are to be had in simple terms in study papers. a few elements of this e-book encompass new examine effects that aren't on hand in different places. as well as routines, over thirty study difficulties are provided during this ebook. during this revised variation nearly each bankruptcy used to be up to date, and a few chapters have been thoroughly rewritten. it's helpful as a textbook for a graduate direction at the topic, in addition to a reference publication for researchers in similar fields. ·

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Consider the following sequence s ~176 of period 7: s ~176 = 01110010111001 ... and the p a t t e r n 0 , 1 9,0, where 9 indicates an arbitrary element. It is easily seen t h a t this p a t t e r n appears only once in a period of the sequence. The notion of a multiplier was first introduced by Carmichael [50]. s) is called a multiplier of the sequence, where s r is the T-shift version of s ~ This multiplier is related to the multiplier of residue difference sets [14]. The T here was called the span of M by Ward [365].

Ca-1 } of Zg, there exists a function f(x) with this partition as its characteristic class. The differential analysis of the system is the analysis of the following difference parameters: df(gi,gj;w) = lCiN(Cj - w ) l , (gi,gj) e G • G, w e ZN. We say that f has the ideal difference property if the values df(gi, gj; w) are approximately the same for all possible (gi, gj;w). To see why the analysis of the difference parameters can be regarded as a kind of differential analysis, we take (G, +) = (Z2, +).

Here Pg (a) could be called the differential of g(x) at a. However, elementary calculus shows that differentials are ideal measures for nonlinearities. Thus, Pg (a) and Pg are nonlinearity measures. The nonlinearity analysis of the cryptographic function f ( x ) refers to the analysis of the probability P r ( f ( x + a) - f ( x ) = b). 18) 2d/(0, 0 ; - a ) = ICol - IC, I + N P r ( f ( x + a) - f ( x ) = 0), 2d/(1, 1 ; - a ) = ]C11- ICol + g P r ( f ( x + a) - f ( x ) = 0), 2d/(1, 0 ; - a ) = 2d/(0, 1 ; - a ) = N - g P r ( f ( x + a) - f ( x ) = 0).

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