# Efficient and secure modular operations using the Adapted Modular Number System

## TL;DR

This paper generalizes the Adapted Modular Number System (AMNS) for efficient modular arithmetic, introducing new algorithms for arithmetic and conversion that enhance speed and security.

## Contribution

It extends AMNS to include polynomials of the form X^n - λ and provides branchless algorithms for arithmetic and conversion operations.

## Key findings

- Multiple AMNS for a given prime p are generated successfully.
- New branchless algorithms improve speed and security.
- The approach generalizes previous AMNS constructions.

## Abstract

The Adapted Modular Number System (AMNS) is a sytem of representation of integers to speed up arithmetic operations modulo a prime p. Such a system can be defined by a tuple (p, n, {\gamma}, {\rho}, E) where E is in Z[X]. In [13] conditions are given to build AMNS with E(X) = {X^n +1}. In this paper, we generalize their results and show how to generate multiple AMNS for a given prime p with E(X)={X^n-\lambda} and {\lambda} in Z. Moreover, we propose a complete set of algorithms without conditional branching to perform arithmetic and conversion operations in the AMNS, using a Montgomery-like method described in [15].