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Large families of pseudorandom binary sequences and lattices are constructed by using the multiplicative inverse and estimates of exponential sums in a finite field. Pseudorandom measures of binary sequences and lattices are studied.

In a series of papers many Boolean functions with good cryptographic properties were constructed using number-theoretic methods. We construct a large family of Boolean functions by using polynomials over finite fields, and study their cryptographic properties: maximum Fourier coefficient, nonlinearity, average sensitivity, sparsity, collision and avalanche effect.

Let $q$, $h$, $a$, $b$ be integers with $q>0$. The classical and the homogeneous Dedekind sums are defined by $$s(h,q)=\sum _{j=1}^q\left(\left(\frac{j}{q}\right)\right)\left(\left(\frac{hj}{q}\right)\right),s(a,b,q)=\sum _{j=1}^q\left(\left(\frac{aj}{q}\right)\right)\left(\left(\frac{bj}{q}\right)\right),$$ respectively, where $$\left(\left(x\right)\right)=\left\{\begin{array}{cc}x-\left[x\right]-\frac{1}{2},\hfill & \text{if}x\text{is}\text{not}\text{an}\text{integer};\hfill \\ 0,\hfill & \text{if}x\text{is}\text{an}\text{integer}.\hfill \end{array}\right.$$ The Knopp identities for the classical and the homogeneous Dedekind sum were the following: $$\begin{array}{c}\sum _{d\mid n}\sum _{r=1}^{d}s\left(\frac{n}{d}a+rq,dq\right)=\sigma \left(n\right)s(a,q),\\ \sum _{d\mid n}\sum _{r_1=1}^d\sum _{r_2=1}^{d}s\left(\frac{n}{d}a+r_{1}q,\frac{n}{d}b+r_{2}q,dq\right)=n\sigma \left(n\right)s(a,b,q),\end{array}$$ where $\sigma \left(n\right)={\sum }_{d\mid n}d$. In this paper generalized homogeneous Hardy sums and Cochrane-Hardy sums are defined, and their arithmetic properties are studied. Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums are given.

A positive integer $n$ is called a square-free number if it is not divisible by a perfect square except $1$. Let $p$ be an odd prime. For $n$ with $(n,p)=1$, the smallest positive integer $f$ such that $n^f\equiv 1(modp)$ is called the exponent of $n$ modulo $p$. If the exponent of $n$ modulo $p$ is $p-1$, then $n$ is called a primitive root mod $p$. Let $A\left(n\right)$ be the characteristic function of the square-free primitive roots modulo $p$. In this paper we study the distribution $$\sum _{n\le x}A\left(n\right)A(n+1),$$ and give an asymptotic formula by using properties of character sums.

The main purpose of the paper is to study, using the analytic method and the property of the Ramanujan’s sum, the computational problem of the mean value of the mixed exponential sums with Dirichlet characters and general Gauss sum. For integers $m$, $n$, $k$, $q$, with $k\ge 1$ and $q\ge 3$, and Dirichlet characters $\chi$, $\overline{\chi }$ modulo $q$ we define a mixed exponential sum $$C(m,n;k;\chi ;\overline{\chi };q)=\sum _{a=1}^q{width0ptheight1em}^{\text{'}}\chi \left(a\right)G_k(a,\overline{\chi })e\left(\frac{m{a}^k+n\overline{a^k}}{q}\right),$$ with Dirichlet character $\chi$ and general Gauss sum $G_k(a,\overline{\chi })$ as coefficient, where ${\sum }^{\text{'}}$ denotes the summation over all $a$ such that $(a,q)=1$, $a\overline{a}\equiv 1modq$ and $e\left(y\right)={\mathrm{e}}^{2\pi \mathrm{i}y}$. We mean value of $$\sum _m\sum _{\chi }\sum _{\overline{\chi }}{\left|C(m,n;k;\chi ;\overline{\chi };q)\right|}^4,$$ and...