Paper 2019/1304
Reduction Modulo $2^{448}-2^{224}-1$
Kaushik Nath and Palash Sarkar
Abstract
An elliptic curve known as Curve448 defined 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. This paper describes efficient algorithms for reduction modulo $p$ that are required for performing field arithmetic in $\mathbb{F}_p$ using 7-limb representation. A key feature of our work is that we provide the relevant proofs of correctness of the algorithms. We also report efficient 64-bit assembly implementations for key generation and shared secret computation phases of the Diffie-Hellman key agreement protocol on Curve448. Timings results on the Haswell and Skylake processors demonstrate that the new 64-bit implementations for computing the shared secret are faster than the previously best known 64-bit implementations.
Note: Minor revision
Metadata
- Available format(s)
-
PDF
- Category
- Public-key cryptography
- Publication info
- Preprint. MINOR revision.
- Keywords
- Curve448Goldilocks primemodulo reductionelliptic curve cryptographyDiffie-Hellman key agreement.
- Contact author(s)
-
kaushikn_r @ isical ac in
palash @ isical ac in - History
- 2022-01-06: last of 3 revisions
- 2019-11-11: received
- See all versions
- Short URL
- https://ia.cr/2019/1304
- License
-
CC BY
BibTeX
@misc{cryptoeprint:2019/1304, author = {Kaushik Nath and Palash Sarkar}, title = {Reduction Modulo $2^{448}-2^{224}-1$}, howpublished = {Cryptology ePrint Archive, Paper 2019/1304}, year = {2019}, note = {\url{https://eprint.iacr.org/2019/1304}}, url = {https://eprint.iacr.org/2019/1304} }