Boolean Functions in Cryptology and Information Security by O.A. Logachev

By O.A. Logachev

This booklet includes the court cases of the NATO-Russia complex learn Institute (ASI) 'Boolean services in Cryptology and data Security', which used to be held in Zvenigorod, Moscow area, Russia. those complaints encompass 3 elements. the 1st half includes survey lectures on quite a few parts of Boolean functionality concept which are of fundamental value for cryptology. those lectures have been brought through top researchers from many nations and include either vintage and up to date effects. the second one half includes examine papers written by means of graduate and postgraduate scholars of Lomonosov college, Moscow.The 3rd half incorporates a record of open difficulties in Boolean functionality idea. The e-book comprises lectures and papers crisis the next parts: cryptographic houses of Boolean services and mappings; algebraic and combinatorial structures of Boolean capabilities and mappings with prescribed cryptographic homes; Boolean capabilities and mappings in cryptosynthesis; class of Boolean services; cryptanalysis of ciphers; and, effective computations in finite fields.

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Additional resources for Boolean Functions in Cryptology and Information Security (Nato Science for Peace and Security)

Sample text

By U (R, R1 ) we denote the set of all functions f : S n → S1 satisfying (2). Consider again the problem: given f , decide whether f ∈ U (R, R1 ). For simplicity, let S = {0, 1, . . , k − 1}, S n be ordered lexicographically, and f be specified by the string of all its values according to this order. Let N = k n . For this problem, also there is an algorithm with complexity O(N 2 ), but it is not that simple to reduce this complexity in general. The next theorem shows how to find a faster algorithm using a simple reduction of this decision problem to evaluating algebraic expression.

Of n-bit words will have then the longest possible period (of length 2n ), and strict uniform distribution; that is, each n-bit word will occur within the period exactly once. In order not to spoil the uniform distribution, one may take a balanced output function F : Z/2n Z → Z/2m Z as a filter. That is, to each m-bit word the mapping F maps the same number of n-bit words (hence; m ≤ n). For m = n balanced mappings are just invertible (that is, bijective, one-to-one) mappings. We say also that the function F on n bit words is bijective modulo 2n whenever the mapping F : Z/2n Z → Z/2n Z is invertible.

The following proposition, whose proof is also based on Theorem 1, gives a method to construct new invertible T -functions (respectively, T -functions with a single cycle property), out of given T -functions: Proposition 1 ([6,7]). Let F be an (n+1)-variate T -function such that for all z1 , . . , zn the T -function F (x, z1 , . . , zn ) is invertible with respect to the variable x. Then the composition F (f (x), 2g1 (x), . . , 2gn (x)) is invertible for arbitrary T -functions g1 , . . , gn and any invertible T -function f .

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