Based on differential analysis in modern cryptography, this paper studies the differential probability of the modular addition operation in the cipher algorithm of ARX structure. This paper first researches characteristics of modulo addition operations through binary expansion and expresses probabilities by carry bit difference expression calculation. By using the Markov chain and state transition matrix, we determine the probability distribution of the number of solutions for a modulo addition difference equation system containing two equations with respect to the difference value, and calculate the mathematical expectation of the number of solutions. At last, this paper provides future research on related probability problems with new insights and methods.

Minghao Bai. A Probabilistic Problem Related to Difference Equations for Modulo Additions. Academic Journal of Mathematical Sciences (2024) Vol. 5, Issue 3: 130-136. https://doi.org/10.25236/AJMS.2024.050314.

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