Analysis of mixing properties of the operations of modular addition and bitwise addition defined on one carrier
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ANALYSIS OF MIXING PROPERTIES OF THE OPERATIONS OF MODULAR ADDITION AND BITWISE ADDITION DEFINED ON ONE CARRIER L. V. Kovalchuka and O. A. Sirenkob
UDC 621.391:519.2:519.7
Abstract. Some results are obtained concerning the influence of bitwise (modular) addition on the structure of the quotient group of a particular subgroup under the operation of modular (bitwise) addition on the set of binary vectors depending on the type of the chosen subgroup. Keywords: residue ring, quotient group, algebraic and statistical attacks, mixing properties of operations. INTRODUCTION One of the modern problems in applied cryptography is creating attack-proof cryptographic primitives that are simple and easily implementable. This raises the problem on finding a set of operations on a set of bit vectors (plain texts) such that, on the one hand, are easily implementable in both software and hardware and, on the other hand, possess “good mixing properties” [1–3]. Alternating operations with such properties makes the primitive stable against various algebraic and statistical attacks, which makes it possible to construct primitives of simple and easy-to-implement structure. The study [1] analyzed the operation of addition (multiplication) in a finite field acting on cosets under multiplication (addition). It was shown that the action of the operation of addition (multiplication) on elements of cosets under the operation of multiplication (addition) essentially destroys the structure of the corresponding quotient group. Based on the obtained results, it was concluded in [1] that applying the composition of these operations to construct an encryption algorithm makes it attack-proof based on homomorphisms [1–3]. However, modern encryption algorithms (for example, [4–6, 8]) much more often use the composition of other operations, namely, of modular and bitwise addition. Therefore, the analysis of mixing properties of group operations of bitwise and modular addition whose carrier is a set of binary vectors is no less important and interesting. Such properties of algebraic operations also characterize the stability of ciphers against attacks of differential cryptoanalysis [7, 8]. We will analyze here the problems similar to those considered in [1, 9] and present the results that characterize the mixing properties of the operations of bitwise and modular addition. We will show that depending on the chosen subgroup in (Vn ,Å ) , the operation of modular addition in (Í2 n , +) can both essentially destroy the structure of the quotient group with respect to the chosen subgroup and completely preserve it. Also, for any subgroup in (Í 2 n , +) the operation of bitwise addition is shown to always preserve the structure of the corresponding quotient group. AUXILIARY NOTATION AND RESULTS The following notation and statements are used to prove the main results. In what follows, by (Vn ,Å ) we will mean a set of vectors of length n with the operation of bitwise addition, and by (Í2 n , +) an additive group of residue ring Í2 n . Let a
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