# Improved Montgomery modular inverse algorithm

@article{McIvor2004ImprovedMM, title={Improved Montgomery modular inverse algorithm}, author={Ciaran McIvor and M{\'a}ire McLoone and John V. McCanny}, journal={Electronics Letters}, year={2004}, volume={40}, pages={1110-1112} }

A new, single and unified Montgomery modular inverse algorithm, which performs both classical and Montgomery modular inversion, is proposed. This reduces the number of Montgomery multiplication operations required by 33% when compared with previous algorithms reported in the literature. The use of this in practice has been investigated by implementation of the improved unified algorithm and the previous algorithms on FPGA devices. The unified algorithm implementation shows a significant speed…

## 24 Citations

Modified Montgomery Modular Inversion with Reduced Number of Multiplications

- Mathematics, Computer ScienceTENCON 2006 - 2006 IEEE Region 10 Conference
- 2006

In this paper, a modified algorithm to compute Montgomery modular Inverse that requires less number of Montgomery modular multiplications compared to the best known methods in literature is proposed.…

Improvement to Montgomery Modular Inverse Algorithm

- Computer ScienceIEEE Transactions on Computers
- 2006

Two modified high radix algorithms are proposed which utilize higher radix to reduce iterations needed without increasing complexity much, thereby accelerating the process.

An Improved Dual Field Modular Inversion Algorithm and VLSI Implementation

- Computer Science2009 First International Conference on Information Science and Engineering
- 2009

An improved dual filed modular inversion algorithm which can support inversion operation in both GF(P) and GF(2^m) is proposed in this paper and is implemented in SMIC 0.18 CMOS standard cell technology.

Scalable VLSI design for fast GF (p) montgomery inverse computation

- Computer Science
- 2006

This paper accelerates a scalable GF(p) Montgomery inversion hardware by modifying the original memory unit to include parallel shifting of all bits which was a task handled by the computing unit.

FPGA realization of fully systolic and parallel architecture of Montgomery multipliers

- Computer Science, Mathematics2011 19thTelecommunications Forum (TELFOR) Proceedings of Papers
- 2011

Modules of multipliers are designed in VHDL with parameters easy to change and implemented in FPGA chip from Xilinx Virtex-4 family.

Fast Algorithm and Hardware Architecture for Modular Inversion in GF(p)

- Computer Science2009 Second International Conference on Intelligent Networks and Intelligent Systems
- 2009

The proposed hardware architecture for modular inversion in GF(p) contains only two adders performing computation parallel, and the synthesized results show that the hardware architecture is more resource-efficient and faster.

Research and Hardware Design of Scalable Dual-Field Montgomery Modular Inversion Algorithm

- Computer Science, Mathematics
- 2011

This paper presents an improved scalable dual-field Montgomery modular inversion algorithm which is based on Kaliski algorithm and shows that the hardware design is superior to other hardware designs in operation time, circuit area and flexibility.

High speed hardware architecture to compute galois fields GF(p) montgomery inversion with scalability features

- Computer Science, MathematicsIET Comput. Digit. Tech.
- 2007

The study included remodeling the entire hardware architecture removing the shifter from the scalable computing part and embedding it in the non-scalable memory unit instead, which resulted in a speedup to the complete inversion process with an area increase due to the new memory shifting unit.

Fast modular multiplication execution in residue number system

- Mathematics, Computer Science2016 IEEE Conference on Quality Management, Transport and Information Security, Information Technologies (IT&MQ&IS)
- 2016

An approximate method is used to find the approximate method a residue from division of a multiplication on the given module, based on Residue Number System, that makes it more applicable for the hardware implementation of the cryptography primitives constructed over a simple finite field.

High Speed Modular Divider Based on GCD Algorithm

- Computer Science, MathematicsICICS
- 2007

The conventional GCD algorithm is extended to radix-4 to increase the efficiency of algorithm the number of comparisons is reduced and the algorithm enables very fast computation of division over GF(2m).

## References

SHOWING 1-4 OF 4 REFERENCES

The Montgomery Modular Inverse-Revisited

- Mathematics, Computer ScienceIEEE Trans. Computers
- 2000

A new definition of the Montgomery inverse is given, and efficient algorithms for computing the classical modular inverse, the Kaliski-Montgomery inverse, and the new Montgomery inverse are introduced.

The Montgomery Inverse and Its Applications

- MathematicsIEEE Trans. Computers
- 1995

The right-shifting binary algorithm for modular inversion is shown naturally to compute the new inverse in fewer operations than the ordinary modular inverse.

Analyzing and comparing Montgomery multiplication algorithms

- Computer Science, MathematicsIEEE 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.

Modular multiplication without trial division

- Mathematics, Computer Science
- 1985

A method for multiplying two integers modulo N while avoiding division by N, a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms.