**New Observation on Division Property**

*Bing Sun and Xin Hai and Wenyu Zhang and Lei Cheng and Zhichao Yang*

**Abstract: **Feistel structure is among the most popular choices for designing ciphers. Recently, 3-round/5-round integral distinguishers for Feistel structures with non-bijective/bijective round functions are presented. At EUROCRYPT 2015, Todo proposed the Division Property to effectively construct integral distinguishers for both Feistel and SPN structures. In this paper, firstly, it is proved that if X, which is a subset of F_2^n, has the division property D_k^n, the number of elements in X is at least 2^k, based on which we can conclude that if a multi-set X has the division property D_n^n, it is in some sense equivalent to either F_2^n or the empty set. Secondly, let d be the algebraic degree of the round function of a Feistel structure. If d\le n-1, the corresponding integral distinguishers are improved as follows: there exists a 3-round integral distinguisher with at most 2^n chosen plaintexts and a 4-round integral distinguisher with at most 2^{2n-2} chosen plaintexts. These results can give new insights to both the division property and Feistel structures.

**Category / Keywords: **Feistel structure, Integral cryptanalysis, Division property

**Original Publication**** (with major differences): **Science China, Information Science

**Date: **received 13 May 2015, last revised 21 Aug 2016

**Contact author: **happy_come at 163 com

**Available format(s): **PDF | BibTeX Citation

**Version: **20160822:030333 (All versions of this report)

**Short URL: **ia.cr/2015/459

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