# The Montgomery Inverse and Its Applications

@article{Kaliski1995TheMI, title={The Montgomery Inverse and Its Applications}, author={Burton S. Kaliski}, journal={IEEE Trans. Computers}, year={1995}, volume={44}, pages={1064-1065} }

The Montgomery inverse of b module a is b/sup -1/2/sup n/ mod a, where n is the number of bits in a. The right-shifting binary algorithm for modular inversion is shown naturally to compute the new inverse in fewer operations than the ordinary modular inverse. The new inverse facilitates recent work by Koc on modular exponentiation and has other applications in cryptography. >

## 186 Citations

### New Algorithm for Classical Modular Inverse

- Mathematics, Computer ScienceCHES
- 2002

The left-shift binary algorithm is shown to naturally calculate the classical modular inverse in fewer operations than the algorithm derived from the Montgomery inverse.

### 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.

### 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.…

### Efficient unified Montgomery inversion with multibit shifting

- Computer Science, Mathematics
- 2005

The authors demonstrate that affine co-ordinate implementation provides a comparable speed to that of projective co-ordinates with careful hardware realisation of existing algorithms for calculating inverses in both fields without utilising special moduli or irreducible polynomials.

### Applications of the Montgomery exponent

- Computer Science, MathematicsInternational Conference on Information Technology: Coding and Computing (ITCC'05) - Volume II
- 2005

This work suggests a new modular exponentiation algorithm that uses one Montgomery multiplication less than the number required with the standard method, and illustrates the potential advantage in performance and code size when known cryptographic applications are modified in a way that MEXP replaces the standard modular exponentation.

### Arithmetic Unit for Computations in GF(p) with the Left-Shifting Multiplicative Inverse Algorithm

- Mathematics, Computer ScienceARCS
- 2013

The hardware architecture of an arithmetic unit intended for computing basic operations over a Galois field GF(p) is presented, and the promising left-shifting algorithm that is based on the extended Euclidean algorithm is used.

### Modular Reduction without Pre-computation for Special Moduli

- Mathematics, Computer Science
- 2010

It is proved that there is no way to remove the pre-computation step in the Barret reduction by way of a contradiction of the derived moduli requirement, and this trick can be applied to currently existing cryptographic systems.

### Hardware implementation of a novel inversion algorithm

- Computer Science, Mathematics2003 46th Midwest Symposium on Circuits and Systems
- 2003

A hardware implementation of inversion algorithms for both binary extension and prime fields is presented, varying slightly from the Montgomery inverse algorithm.

### New hardware algorithms and designs for montgomery modular inverse computation in galois fields gf(p) and gf(2n)

- Computer Science, Mathematics
- 2003

This work investigates the GF( p) inversion and presents several phases in the design of efficient hardware implementations to compute the Montgomery modular inverse, and proposes a scalable and unified architecture for a Montgomery inverse hardware that operates in both GF(p) and GF(2n) fields.

### Another Look at Inversions over Binary Fields

- Computer Science, Mathematics2013 IEEE 21st Symposium on Computer Arithmetic
- 2013

New algorithms for one of the most common operations in public key cryptosystems: the inversion over binary Galois fields are offered, which are provably more economical-in terms of the average number of multiplications-than the popular Itoh-Tsujii algorithm.

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