Paper 2019/1304

Reduction Modulo $2^{448}-2^{224}-1$

Kaushik Nath and Palash Sarkar

Abstract

An elliptic curve known as Curve448 over the finite field $\mathbb{F}_p$, where $p=2^{448}-2^{224}-1$ has been proposed as part of the Transport Layer Security (TLS) protocol, version 1.3. Elements of $\mathbb{F}_p$ can be represented using 7 limbs where each limb is a 64-bit quantity. In this paper, we describe efficient algorithms for reduction modulo $p$ that are required for performing field arithmetic in $\mathbb{F}_p$. A key feature of our algorithms is that we provide the relevant proofs of correctness. Based on the proofs of correctness we point out the incompleteness of the reduction methods in the previously known fastest code for implementing arithmetic in $\mathbb{F}_p$.