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G-perfect nonlinear functions. (English) Zbl 1179.94060
Summary: Perfect nonlinear functions are used to construct DES-like cryptosystems that are resistant to differential attacks. We present generalized DES-like cryptosystems where the XOR operation is replaced by a general group action. The new cryptosystems, when combined with $$G$$-perfect nonlinear functions (similar to classical perfect nonlinear functions with one XOR replaced by a general group action), allow us to construct systems resistant to modified differential attacks. The more general setting enables robust cryptosystems with parameters that would not be possible in the classical setting. We construct several examples of $$G$$-perfect nonlinear functions, both $${\mathbb{Z}}_2$$ -valued and $${\mathbb{Z}}_2^a$$ -valued. Our final constructions demonstrate $$G$$-perfect nonlinear planar permutations (from $${\mathbb{Z}}_2^a$$ to itself), thus providing an alternative implementation to current uses of almost perfect nonlinear functions.

##### MSC:
 94A60 Cryptography 05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) 11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
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