# Fast Modular Reduction

@article{Hasenplaugh2007FastMR, title={Fast Modular Reduction}, author={William Hasenplaugh and G. Gaubatz and Vinodh Gopal}, journal={18th IEEE Symposium on Computer Arithmetic (ARITH '07)}, year={2007}, pages={225-229} }

It is widely acknowledged that efficient modular multiplication is a key to high-performance implementation of public-key cryptography, be it classical RSA, Diffie-Hellman, or (hyper-) elliptic curve algorithms. In the recent decade, practitioners have relied mainly on two popular methods: Montgomery Multiplication and regular long-integer multiplication in combination with Barrett's modular reduction technique. In this paper, we propose a modification to Barrett's algorithm that leads to a… Expand

#### 38 Citations

Fast modular reduction for large-integer multiplication for cryptosystem application

- Mathematics, Computer Science
- 2012 Second International Conference on Digital Information and Communication Technology and it's Applications (DICTAP)
- 2012

This paper proposes a revision to the Modified Barrett algorithm based on the properties of Mersenne and Quasi-Mersenne primes, and describes four distinct sets of moduli which are responsible for converting the single-precision multiplication prevalent in many of today's techniques into an addition operation and a few simple shift operations. Expand

Partially Interleaved Modular Karatsuba-Ofman Multiplication

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- 2015

Summary We describe a method of performing modular multiplication that has various applications in the field of modern cryptography and coding theory. The proposed algorithm, which combines the… Expand

Fast Modular Reduction for Large-Integer Multiplication

- Computer Science
- 2012

The work contained in this thesis is a representation of the successful attempt to speed-up the modular reduction as an independent step of modular multiplication, which is the central operation in… Expand

Hardware Implementation of Barrett Reduction Exploiting Constant Multiplication

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- 2019

This thesis studies the influence using constant multipliers has on four different Barrett reduction variants targeting the Virtex-7 and creates a hybrid constant multiplier circuit based on Karatsuba multiplication which uses smaller FloPoCo-generated base multipliers. Expand

Arithmetic Considerations for Isogeny-Based Cryptography

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- IEEE Transactions on Computers
- 2019

In this paper we investigate various arithmetic techniques which can be used to potentially enhance the performance in the supersingular isogeny Diffie-Hellman (SIDH) key-exchange protocol which is… Expand

Fast Arithmetic Modulo 2^x p^y ± 1

- Mathematics, Computer Science
- 2017 IEEE 24th Symposium on Computer Arithmetic (ARITH)
- 2017

An overview of techniques to compute arithmetic modulo 2x py ± 1 and shows that in the SIDH setting, where arithmetic over a quadratic extension field is required, the approaches based on Montgomery multiplication are to be preferred. Expand

An application of Miller Moreno groups to establishment protocol Non commutative cryptography

- Computer Science
- 2020 Fourth World Conference on Smart Trends in Systems, Security and Sustainability (WorldS4)
- 2020

An algorithm of generating (designing) of common key for a non-commutative group with two mutually commuting subgroups has been constructed by us. Expand

Justification of Montgomery Modular Reduction

- Computer Science
- 2012

one of the most known and widely used methods in Cryptography is the method suggested by Peter Montgomery; this method is based on the changing of the original reduction modulo by some other… Expand

Gpu and asic acceleration of elliptic curve scalar point multiplication

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- 2012

As public information is increasingly communicated across public networks such as the internet, the use of public key cryptography to provide security services such as authentication, data integrity,… Expand

Efficient Software-Implementation of Finite Fields with Applications to Cryptography

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- 2006

In this work, we present a survey of efficient techniques for software implementation of finite field arithmetic especially suitable for cryptographic applications. We discuss different algorithms… Expand

#### References

SHOWING 1-10 OF 12 REFERENCES

Analyzing and comparing Montgomery multiplication algorithms

- Computer Science
- IEEE Micro
- 1996

The operations involved in computing the Montgomery product are studied, several high-speed, space-efficient algorithms for computing MonPro(a, b), and their time and space requirements are described. Expand

Implementing the Rivest Shamir and Adleman Public Key Encryption Algorithm on a Standard Digital Signal Processor

- Computer Science
- CRYPTO
- 1986

A description of the techniques employed at Oxford University to obtain a high speed implementation of the RSA encryption algorithm on an "off-the-shelf" digital signal processing chip and the techniques of algorithm development employed lead to a provably correct implementation. Expand

Duality between Multiplication and Modular Reduction

- Mathematics, Computer Science
- IACR Cryptol. ePrint Arch.
- 2005

It is proved that this optimal multiplier recoding technique naturally translates into a canonical modular reduction technique, and the resulting reduction algorithm is optimal with respect to its average-time complexity as well. Expand

The RSA Cryptography Processor

- Computer Science
- EUROCRYPT
- 1987

In commercial applications, a minimum ciphering rate of 64 K bit/sec is required which will be the transmission rate of public digital networks and a single-chip implementation of the RSA algorithm seems to be the only solution. Expand

Modular multiplication without trial division

- Mathematics
- 1985

Let N > 1. We present a method for multiplying two integers (called N-residues) modulo N while avoiding division by N. N-residues are represented in a nonstandard way, so this method is useful only… Expand

A Scalable Architecture for Montgomery Multiplication

- Computer Science
- CHES
- 1999

The general view of the new architecture is described, hardware organization for its parallel computation is analyzed, and design tradeoffs which are useful to identify the best hardware configuration are discussed. Expand

Schnelle Multiplikation großer Zahlen

- Computer Science
- Computing
- 2005

ZusammenfassungEs wird ein Algorithmus zur Berechnung des Produktes von zweiN-stelligen Dualzahlen angegeben. Zwei Arten der Realisierung werden betrachtet: Turingmaschinen mit mehreren Bändern und… Expand

Design of an efficient public-key cryptographic library for RISC-based smart cards

- Ph.D. Dissertation, Université Catholique de Louvain, May
- 1998

Encoding system according to the socalled RSA-method, by means of a microcontroller and arrangement implementing this system

- U.S. Patent #5,166,978,
- 1992