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Preimage distributions of a class of perfect nonlinear functions. (Chinese. English summary) Zbl 1240.94061

Summary: By using the algebraic number theory, the preimage distributions of a class of perfect nonlinear functions are discussed, the functions map an Abelian group of order \(3l\) to an Abelian group of order 3. The integral roots of the equivalent equation \(x^2+xy+y^2=l\) are investigated, and a necessary condition for the existence of this kind of functions and the enumeration of preimage distributions are given. Furthermore, an algorithm which can be used to give all possible preimage distributions of this kind of functions is presented.

MSC:

94A60 Cryptography
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
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