The Mean Distribution Of The Arithmetic Function And Its Application In Algorithm

Arithmetical function is a function whose domain is a set of positive integer and its codomain is a set of complex number.In the field of mathematics,number theory is a branch of mathematics,and arithmetical function is an important part of number theory.Since many number theory problems and combinatorial mathematical problems are to be solved by transforming them into arithmetical functions,this requires our in-depth study of arithmetical functions.However,the value of many arithmetical functions is often very irregular,but research on their mean value can get a beautiful asymptotic formula,so the discussion of the mean of the function has extraordinary significance in the study of arithmetical functions.With the advent of the information age,the study of arithmetical functions is not limited to the field of mathematics.In recent years,the research of arithmetical functions has been widely applied to many aspects such as cryptography,communication technology,computer science and technology.In this paper,we use elementary and analytic methods to study the mean value of Smarandache multiplicative function and the upper bound of exponential sums and characteristic sums,and obtain several kinds of mean value of Smarandache multiplicative function.We also use the upper bound of exponential sums and characteristic sums of low Hamming weight number to prove the uniform distribution property of low Hamming weight which is widely used in the cipher algorithm.The specific research contcents and results are as follows:(1)By using elementary and analytic methods,the mean value of Smarandache multiplicative function S(n)and its reciprocal on simple number sequence,the hybrid mean value of Smarandache multiplicative function S(n)and the maximum prime factor function on M-power complement are studied,and a new Smarandache multiplication function is constructed and a kind of mean value of this function is calculated.(2)By using the properties of exponential sums and Erdos-Turan inequality,the upper bound of a kind of exponential sums on the low Hamming weight number is calculated,and the uniform distribution property of the low Hamming weight sequence is proved by using the upper bound and the discrepancy of the sequence,ensuring the pseudo-randomness and operation efficiency of the low Hamming weight sequence in the encryption algorithm.(3)It is proved that the low Hamming weight product(LHWP)X=x1x2x3(where xi(i=1,2,3)have a smaller Hamming weight)is ?-statistically nearly uniformly distributed by using analytic method and the properties of characteristic sums.That is to say,the LHWP algorithm is pseudorandom,thus ensuring the security of the low Hamming weight product in the construction of cryptographic algorithm.