On perfect hashing of numbers with sparse digit representation via multiplication by a constant

Digit Preserving Multiplication in Continued Fraction Representations

A permutiple is a number which is an integer multiple of some permutation of its digits. A well-known example is 9801 since it is an integer multiple of its reversal, 1089. In this paper, we consider the permutiple problem in an entirely different setting: continued fractions. We pose the question of when the simple continued fraction representation of a rational number is an integer multiple of a permutation of its partial quotients (or digits, as we shall call them). We develop some general results and apply them to finding new examples. In doing so, we attempt to classify all 2, 3, and 4-di…

The Plum-Blossom Wedge Product Method for Multiplication and Division of Multi-digit Integers

For two integers $a$ and $b$, the plum-blossom product of $a$ and $b$ is defined as the ones of the usual product of $a$ and $b$ if the ones is less than or equal to $3$, or otherwise as the ones minus $10$. Under this operation, all integers is a commutative semigroup. Since some fine properties of this semigroup, the plum-blossom product method is the efficient method of multiplication and division of long integers, can be used to mental arithmetic and to computer science. In this paper, we improve the theory of plum-blossom products so that the new concept--the plum-blossom wedge product is…

使用对乘法封闭的数字集的高斯整数与爱森斯坦整数的最优表示

Optimal Representations of Gaussian and Eisenstein Integers using digit sets closed under multiplication

本文研究高斯整数和爱森斯坦整数在特定基与数字集下的最优表示，给出最优表示个数的上界并刻画达到上界的整数，推广了有符号二进制记数系统的相关结果。

On Permutiples Having a Fixed Set of Digits

A permutiple is the product of a digit preserving multiplication, that is, a number which is an integer multiple of some permutation of its digits. Certain permutiple problems, particularly transposable, cyclic, and, more recently, palintiple numbers, have been well-studied. In this paper we study the problem of general digit preserving multiplication. We show how the digits and carries of a permutiple are related and utilize these relationships to develop methods for finding new permutiple examples from old. In particular, we shall focus on the problem of finding new permutiples from a known …

Arithmetics in number systems with cubic base

This paper focuses on greedy expansions, one possible representation of numbers, and on arithmetical operations with them. Performing addition or multiplication some additional digits can appear. We study bounds on the number of such digits assuming the finiteness of the expansion of the considered sum or product, especially for the case of cubic Pisot units.

Speeding Up Elliptic Curve Multiplication with Mixed-base Representation for Applications to SIDH Ciphers

Elliptic curve multiplications can be improved by replacing the standard ladder algorithm's base 2 representation of the scalar multiplicand, with mixed-base representations with power-of-2 bases, processing the n bits of the current digit in one optimized step. For this purpose, we also present a new methodology to compute, for Weierstrass form elliptic curves in the affine plane, operations of the type mP+nQ where m and n are small integers. This provides implementations with the lower cost than previous algorithms, using only one inversion. In particular, the proposed techniques enable more…