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

By O.A. Logachev

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

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Extra resources for Boolean Functions in Cryptology and Information Security

Sample text

Al. studied the so-called determined functions since 1950-th, see [2]. Actually, a determined function is a function that could be represented by an automaton. Consider an automaton with a binary input and binary output; then each infinite input string of 0s and 1s the automaton transforms into an infinite output string of 0s and 1s (we suppose that the initial state is fixed). Note that every outputted i-th bit depends only on the inputted i-th bit and on the current state of the automaton. Yet the current state depends only on the previous state and on the (i − 1)-th input bit.

Thus the length of all integers in this algorithm is O(log N ) and the number of bit operations is O(N log N ) · O(log2 N ) = O(N log3 N ). In the proof of the above theorem we used the logical semiring (0, N) and calculations were in its extension, namely, the set of integers. In the next example we use the same logical semiring (0, N), but calculations are over the complex numbers. Consider the set P2∗ of all partial Boolean functions (“partial” means that function can take values 0, 1 or ∗ (undefined)).

Consider the following relations over E2 : R1 (y) ≡ (y = 0); R2 (y) ≡ (y = 1); R3 (y1 , y2 ) ≡ (y1 = 0) & (y2 = 1); R4 (y1 , y2 ) ≡ (y1 ≤ y2 ); R5 (y1 , y2 ) ≡ (y1 = y2 ); R6 (y1 , y2 , y3 , y4 ) ≡ y1 + y2 + y3 + y4 = 0 (mod 2); R7 (y1 , y2 , y3 , y4 ) ≡ (y1 = y2 ) & (y3 = y4 ) ∨ (y1 = y3 ) & (y2 = y4 ). Let also ∗ denote nowhere defined function. Theorem 3 (Freivald [3]). A set A of partial Boolean functions is complete if and only if it does not belong to any of the following 8 classes: P2 ∪{∗}, U (Ri , Ri∗ ), i = 1, 2, .

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