By O.A. Logachev

ISBN-10: 1435678125

ISBN-13: 9781435678125

ISBN-10: 1586038788

ISBN-13: 9781586038786

This booklet comprises the complaints of the NATO-Russia complex examine Institute (ASI) 'Boolean services in Cryptology and knowledge Security', which was once held in Zvenigorod, Moscow sector, Russia. those court cases include 3 elements. the 1st half comprises survey lectures on a number of parts of Boolean functionality idea which are of basic value for cryptology. those lectures have been introduced through top researchers from many nations and include either vintage and up to date effects. the second one half comprises learn 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 concept. The ebook comprises lectures and papers situation the next components: cryptographic homes of Boolean features and mappings; algebraic and combinatorial buildings of Boolean services and mappings with prescribed cryptographic homes; Boolean capabilities and mappings in cryptosynthesis; type of Boolean features; cryptanalysis of ciphers; and, effective computations in finite fields.

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**Extra info for Boolean Functions in Cryptology and Information Security **

**Sample text**

Consider the set U (R, R1 ), where R1 (y1 , y2 , y3 ) is arbitrary relation over S1 and R(y1 , y2 , y3 ) is the following relation over S: R(y1 , y2 , y3 ) ≡ (y1 + y2 + y3 ≥ 2). Algorithm for deciding whether f ∈ U (R, R1 ) based on implication (4) has complexity of order at least N 3 where N = 2n and n is the number of variables. Theorem 5. Let f (x1 , . . , xn ) be speciﬁed by the string of all its values. Then there exists an algorithm with bit complexity O(N log5 N ) for deciding whether f ∈ U (R, R1 ).

J R(αj , βj , . . , δj ). If R1 (y1 , . . , ym ) is any relation over S1 then deﬁne the set U (R, R1 ) to be the set of all functions f : S n → S1 satisfying the implication Rn (α1 , . . , αm ) =⇒ R1 (f (α1 ), . . , f (αm )) (4) for any α1 , . . , αm in S n . ) Consider now the set U (R, R1 ) where R1 (y1 , y2 , y3 , y4 ) is arbitrary relation over S1 and R(y1 , y2 , y3 , y4 ) is the next relation over S: R(y1 , y2 , y3 , y4 ) ≡ (y1 + y2 + y3 + y4 = 0 (mod 2)). V. ) Algorithm for deciding whether f ∈ U (R, R1 ) based on implication (4) has complexity of order at least N 4 , where N = 2n and n is the number of variables.

Thus if we denote (α1 , . . , αn−1 ) by α0 , (β1 , . . , βn−1 ) by β0 , (γ1 , . . , γn−1 ) by γ0 , and (δ1 , . . , δn−1 ) by δ0 , then we can transform T over Z as follows: 28 V. Alekseev / On Algebraic Algorithms Deciding Properties of Discrete Functions t(α1 , β1 , γ1 , δ1 ) · · · t(αn−1 , βn−1 , γn−1 , δn−1 ) T = α0 β0 γ0 δ0 1 1 1 1 × t(αn , βn , γn , δn )xα0 ,αn yβ0 ,βn zγ0 ,γn vδ0 ,δn αn =0 βn =0 γn =0 δn =0 t(α1 , β1 , γ1 , δ1 ) · · · t(αn−1 , βn−1 , γn−1 , δn−1 ) = α0 β0 γ0 δ0 × (xα0 ,0 + xα0 ,1 ) yβ0 ,0 + yβ0 ,1 (zγ0 ,0 + zγ0 ,1 ) vδ0 ,0 + vδ0 ,1 + (xα0 ,0 − xα0 ,1 ) yβ0 ,0 − yβ0 ,1 (zγ0 ,0 − zγ0 ,1 ) vδ0 ,0 − vδ0 ,1 .

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

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