Welcome to the resource topic for
**2023/911**

**Title:**

General Results of Linear Approximations over Finite Abelian Groups

**Authors:**
Zhongfeng Niu, Siwei Sun, Hailun Yan, Qi Wang

**Abstract:**

In recent years, progress in practical applications of secure multi-party computation (MPC), fully homomorphic encryption (FHE), and zero-knowledge proofs (ZK) motivate people to explore symmetric-key cryptographic algorithms, as well as corresponding cryptanalysis techniques (such as differential cryptanalysis, linear cryptanalysis), over general finite fields \mathbb{F} or the additive group induced by \mathbb{F}^n. This investigation leads to the break of some MPC/FHE/ZK-friendly symmetric-key primitives, the United States format-preserving encryption standard FF3-1 and the South-Korean standards FEA-1 and FEA-2. In this paper, we revisit linear cryptanalysis and give general results of linear approximations over arbitrary finite Abelian groups. We consider the nonlinearity, which is the maximal non-trivial linear approximation, to characterize the resistance of a function against linear cryptanalysis. The lower bound of the nonlinearity of a function F:G\rightarrow H over an arbitrary finite Abelian group was first given by Pott in 2004. However, the result was restricted to the case that the size of G divides the size of H due to its connection to relative difference sets. We complete the generalization from \mathbb{F}_2^n to finite Abelian groups and give the lower bound of \lambda_F for all different cases. Our result is deduced by the new links that we established between linear cryptanalysis and differential cryptanalysis over general finite Abelian groups.

**ePrint:**
https://eprint.iacr.org/2023/911

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