Automatic Search of Bit-Based Division Property for ARX Ciphers and Word-Based Division Property

  • Ling Sun
  • Wei Wang
  • Meiqin WangEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10624)


Division property is a generalized integral property proposed by Todo at Eurocrypt 2015. Previous tools for automatic searching are mainly based on the Mixed Integer Linear Programming (MILP) method and trace the division property propagation at the bit level. In this paper, we propose automatic tools to detect ARX ciphers’ division property at the bit level and some specific ciphers’ division property at the word level.

For ARX ciphers, we construct the automatic searching tool relying on Boolean Satisfiability Problem (SAT) instead of MILP, since SAT method is more suitable in the search of ARX ciphers’ differential/linear characteristics. The propagation of division property is translated into a system of logical equations in Conjunctive Normal Form (CNF). Some logical equations can be dynamically adjusted according to different initial division properties and stopping rule, while the others corresponding to r-round propagations remain the same. Moreover, our approach can efficiently identify some optimized distinguishers with lower data complexity. As a result, we obtain a 17-round distinguisher for SHACAL-2, which gains four more rounds than previous work, and an 8-round distinguisher for LEA, which covers one more round than the former one.

For word-based division property, we develop the automatic search based on Satisfiability Modulo Theories (SMT), which is a generalization of SAT. We model division property propagations of basic operations and S-boxes by logical formulas, and turn the searching problem into an SMT problem. With some available solvers, we achieve some new distinguishers. For CLEFIA, 10-round distinguishers are obtained, which cover one more round than the previous work. For the internal block cipher of Whirlpool, the data complexities of 4/5-round distinguishers are improved. For Rijndael-192 and Rijndael-256, 6-round distinguishers are presented, which attain two more rounds than the published ones. Besides, the integral attacks for CLEFIA are improved by one round with the newly obtained distinguishers.


Automatic search Division property ARX SAT/SMT 



The authors would like to thank the anonymous reviewers of Asiacrypt 2017 for their helpful comments. This work was supported by the 973 Program (No. 2013CB834205), NSFC Projects (No. 61572293), Science and Technology on Communication Security Laboratory of China (No. 9140c110207150c11050), as well as Chinese Major Program of National Cryptography Development Foundation (No. MMJJ20170102).

Supplementary material


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Copyright information

© International Association for Cryptologic Research 2017

Authors and Affiliations

  1. 1.Key Laboratory of Cryptologic Technology and Information Security, Ministry of EducationShandong UniversityJinanChina
  2. 2.Science and Technology on Communication Security LaboratoryChengduChina
  3. 3.State Key Laboratory of CryptologyBeijingChina

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