Paper 2026/775

Differential and Linear Cryptanalysis of Modular Addition

Ali Doğan, Istanbul Settlement and Custody Bank

Gökçe Yetişer

Abstract

This paper presents a comprehensive analysis of modular addition from a cryptanalytic perspective, focusing on both linear and differential cryptanalysis techniques. We examine the probability distribution of carry bits in modular addition operations and demonstrate how these probabilities affect linear approximations. The paper provides detailed algorithms for constructing Linear Approximation Tables (LAT) and Difference Distribution Tables (DDT) for modular addition operations, along with theoretical proofs and practical examples. Our analysis reveals that the probability of carry bits approaches 1/2 as the bit position increases, which significantly impacts the effectiveness of linear cryptanalysis. Furthermore, we demonstrate how to extend DDTs for larger bit sizes by leveraging smaller tables and carry bit relationships. The findings have direct implications for the cryptanalysis of ARX ciphers.

@misc{cryptoeprint:2026/775,
      author = {Halil İbrahim Kaplan and Ali Doğan and Gökçe Yetişer},
      title = {Differential and Linear Cryptanalysis of Modular Addition},
      howpublished = {Cryptology {ePrint} Archive, Paper 2026/775},
      year = {2026},
      url = {https://eprint.iacr.org/2026/775}
}