Analysis of mixing properties of the operations of modular addition and bitwise addition defined on one carrier

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.

Mixing Properties of Operations Defined on the Set of N-Dimensional Vectors Over a Prime Finite Field

“…In the present paper, we will generalize the results from [3,4] for bitwise and modular addition and will show that depending on the choice of subgroup in V p n ( ), addition in Z p n can both considerably destroy the structure of the quotient group of the chosen subgroup and completely retain it. We will also show that for any subgroup in Z p n , bitwise addition always retains the structure of the respective quotient group.…”

“…In [3,4], conclusions are made that characterize the mixing properties of operations of bitwise and modular addition specified on the same carrier.…”

“…In [5], the results are obtained that concern nontrivial quotient structures available for certain transformations of additive groups, in particular, the result from [3,4] is generalized, which concerns the existence of invariant subgroups of the additive group of ring Z n 2 and subspaces of linear space V n . However, these studies only prove the existence of invariant subgroups, but do not determine the probabilities of the membership in various cosets of the sum of elements belonging to cosets with respect to non-normal subgroup.…”

“…z i = 1.In the case b 1 0 = : 1 and u 1 1 = .From formula(3) it follows that the number of cosets with respect to the subgroup G to which the difference of elements c and d modulo p n belongs with nonzero probability is2 k if b 1 0 > and 2 1 k -if b 1 0 = since each E i (except for E 1 )can take two values; E 1 takes one value if b 1 0 = and two values if b 1 0 ¹ . 3.…”