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旋转对称函数可以极大地提高密码算法的运算效率,节省资源开销,因此在密码学与编码理论中有着广泛的应用.关于旋转对称函数的计数问题一直是该领域研究的重点问题.Li等人将旋转对称布尔函数的概念推广到素域GF(p)上,给出了GF(p)上平衡的旋转对称布尔函数个数的下界,并将次数大于3的齐次旋转对称函数的计数问题作为一个公开的难题.本文进一步研究了这个公开问题,将其转化为对极小旋转对称函数的计数,证明了极小旋转对称函数与GF(p)n中的轨道是一一对应的.然后利用容斥原理和莫比乌斯变换,得到了代数次数任意的极小旋转对称函数的计数公式,最后给出了GF(p)上齐次旋转对称函数的计数公式.与已有的结果相比,该公式具有简单性、统一性等特点.
Rotationally symmetric functions can greatly improve the computational efficiency of cryptographic algorithms and save resource overhead, so they have been widely used in cryptography and coding theory. Counting of rotational symmetric functions has been the key issue in this field. The concept of rotationally symmetric boolean functions is generalized to the prime field GF (p), and the lower bound of the number of symmetric Boolean functions balanced on GF (p) is given, and the counting problem of homogeneous rotational symmetric functions whose number is greater than 3 is taken as An open problem.This paper further investigates this open problem and transforms it into the counting of minimal rotational symmetry functions, and proves that the symmetric function of minimal rotation is one-to-one correspondence with the orbits in GF (p) n. Then, The principle of exclusion and the Mobius transformation, we obtain the counting formula of any minimal rotational symmetry function with algebraic number, and finally give the counting formula of the homogeneous rotational symmetry function on GF (p). Compared with the existing results , The formula has the characteristics of simplicity, unity and so on.