Binary Signed-Digit Integers and the Stern Polynomial

TL;DR

This paper uncovers new connections between binary signed-digit representations and Stern polynomials, providing recursive formulas and an efficient algorithm for analyzing BSD representations of integers.

Contribution

It introduces novel links between BSD representations and Stern polynomials, including new recursions and a linear-time algorithm for computing BSD properties.

Findings

Derived a weight-distribution theorem for BSD representations using Stern polynomial coefficients.

Established new recursive relations for Stern polynomials and BSD representations.

Developed an $ ext{O}(n)$ algorithm to compute BSD representation characteristics for all integers up to n.

Abstract

The binary signed-digit representation of integers is used for efficient computation in various settings. The Stern polynomial is a polynomial extension of the well-studied Stern diatomic sequence, and has itself has been investigated in some depth. In this paper, we show previously unknown connections between BSD representations and the Stern polynomial. We derive a weight-distribution theorem for $i$-bit BSD representations of an integer $n$ in terms of the coefficients and degrees of the terms of the Stern polynomial of $2^i-n$. We then show new recursions on Stern polynomials, and from these and the weight-distribution theorem obtain similar BSD recursions and a fast $\mathcal{O}(n)$ algorithm that calculates the number and number of $0$s of the optimal BSD representations of all of the integers of NAF-bitlength $\log(n)$ at once, which then may be compared.