Abstract
This paper investigates the behavior of rotating binaries. A rotation by r digits to the left of a binary number B exhibits in particular cases the divisibility l∣N1(B)·r+1, where l is the bit-length of B and N1(B) is the Hamming weight of B, that is the number of ones in B. The integer r is called the left-rotational distance. We investigate the connection between this rotational distance, the length, and the Hamming weight of binary numbers. Moreover, we follow the question under which circumstances the above-mentioned divisibility is true. We have found out and will demonstrate that this divisibility occurs for kn+c cycles.
Highlights
With this manuscript, we pursue the goal of exploring the connection between rotations of binary vectors and 3n + c cycles
This investigation is motivated by the use case of information encryption and efficiency improvement of cryptographic algorithms, especially of those algorithms that are implemented by a linear feedback shifting register (LFSR) as demonstrated by Grosek and Hromada [3]
The divisibility feature l | N1(B) · r + 1, demonstrated in Section 3 holds for the generalized kn + c cycles
Summary
We pursue the goal of exploring the connection between rotations of binary vectors and 3n + c cycles. In the following we will develop a computational base for the binary rotation, its related cycles and generalize the divisibility feature. Let us take a binary number B of length l with N1(B) ones (and N0(B) = l − N1(B) zeros), for example l = 8, N1(B) = 5 and B = 10110101 = 181, the minimum that is obtainable by rotating B is Bmin = 01011011 = 91 and the maximum is Bmax = 11011010 = 218. Based on the fact that binary rotations lead to 3n + c cycles, Section 3 contributes as well to the question under which conditions the divisibility is granted. The generalization of binary rotations to kn + c cycles in Section 9 expands the field of the investigated divisibility behavior
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