Takashi Horiyama, Yasuhide Numata, Kazuhisa Seto et al.
In this article, we consider the words with cyclic indices. For given $s$, we consider the pair $(ι,κ)$ of indices such that the word of length $s$ from $ι$ is equal to the word of length $s$ from $κ$. We give a characterization of such pairs for a cyclic Fibonacci word, and give the number of them.
Shizuo Kaji, Toshiaki Maeno, Koji Nuida et al.
It is known that any $n$-variable function on a finite prime field of characteristic $p$ can be expressed as a polynomial over the same field with at most $p^n$ monomials. However, it is not obvious to determine the polynomial for a given concrete function. In this paper, we study the concrete polynomial expressions of the carries in addition and multiplication of $p$-ary integers. For the case of addition, our result gives a new family of symmetric polynomials, which generalizes the known result for the binary case $p = 2$ where the carries are given by elementary symmetric polynomials. On the other hand, for the case of multiplication of $n$ single-digit integers, we give a simple formula of the polynomial expression for the carry to the next digit using the Bernoulli numbers, and show that it has only $(n+1)(p-1)/2 + 1$ monomials, which is significantly fewer than the worst-case number $p^n$ of monomials for general functions. We also discuss applications of our results to cryptographic computation on encrypted data.
Koji Nuida, Takuro Abe, Shizuo Kaji et al.
In this paper, we specify a class of mathematical problems, which we refer to as "Function Density Problems" (FDPs, in short), and point out novel connections of FDPs to the following two cryptographic topics; theoretical security evaluations of keyless hash functions (such as SHA-1), and constructions of provably secure pseudorandom generators (PRGs) with some enhanced security property introduced by Dubrov and Ishai [STOC 2006]. Our argument aims at proposing new theoretical frameworks for these topics (especially for the former) based on FDPs, rather than providing some concrete and practical results on the topics. We also give some examples of mathematical discussions on FDPs, which would be of independent interest from mathematical viewpoints. Finally, we discuss possible directions of future research on other cryptographic applications of FDPs and on mathematical studies on FDPs themselves.