Research on the Security Criterion of S-Boxes Against Division Property

As a generalized integral property, division property was proposed by Todo at EUROCRYPT 2015. We propose a new security criterion of S-boxes against division property and prove that it is invariant under permutation-xor equivalence. Based on the criterion, the division properties of some important 4-bit S-boxes are showed. Then, we apply it to improve the resistance of ciphers against division-property-based integral attacks while keeping the same security level against other attacks. Specifically, the resistance of the cipher PRESENT against division-property-based integral attack is improved by 2 rounds, and the resistance of the cipher LBlock against division-property-based integral attack is improved by 1 round.