# Modular multiplication without trial division

@article{Montgomery1985ModularMW, title={Modular multiplication without trial division}, author={Peter L. Montgomery}, journal={Mathematics of Computation}, year={1985}, volume={44}, pages={519-521} }

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 if several computations are done modulo one N. The addition and subtraction algorithms are unchanged. 1. Description. Some algorithms (1), (2), (4), (5) require extensive modular arith- metic. We propose a representation of residue classes so as to speed modular multiplication without affecting the…

## 2,574 Citations

Some improvements on RNS Montgomery modular multiplication

- Computer Science, MathematicsSPIE Optics + Photonics
- 2000

An algorithmic parallel algorithm is proposed for this translation from RNS to Mixed Radix, using a result that comes from an RNS division algorithm, and obtaining in a logarithmic time an approximation of the Mixed radix representation.

Modular multiplication and base extensions in residue number systems

- Computer Science, MathematicsProceedings 15th IEEE Symposium on Computer Arithmetic. ARITH-15 2001
- 2001

A new RNS modular multiplication for very large operands is presented, based on Montgomery's (1985) method adapted to residue arithmetic, which achieves an effect corresponding to a redundant high-radix implementation by choosing the moduli of the RNS system reasonably large.

Montgomery Reduction with Even Modulus

- Computer Science, Mathematics

This short paper shows that, with the help of the Chinese Remainder Theorem, the Mont-gomery reduction algorithm can be used to eeciently perform these modular arithmetic operations with respect to an even modulus.

Multiplication of large integers by the use of modular arithmetic: application to cryptography

- Computer Science, MathematicsSGSC
- 1990

Choice for modulil are made to compute the Inverse modulo efficiently without a need for the Euclid's algorithm and the principles of modular arithmetic and the Chinese remainder theorem, with efficient methods are given in detail.

A VLSI algorithm for modular multiplication/division

- Computer Science, MathematicsProceedings 2003 16th IEEE Symposium on Computer Arithmetic
- 2003

The algorithm is based on Montgomery's method for modular multiplication and on the extended binary GCD algorithm for modular division and carries out an n-bit modular multiplication in at most 2n+5 clock cycles, where the length of the clock cycle is constant and independent of n.

Recursive Double-Size Modular Multiplications from Euclidean and Montgomery Multipliers

- Computer Science, MathematicsIEICE Trans. Fundam. Electron. Commun. Comput. Sci.
- 2010

This paper addresses the computation cost and improves on previous 2l-bit modular multiplication algorithms to return not only the remainder but also the quotient, resulting in an higher performance in the recursive approach, which becomes twice faster in the quadrupling case and four times faster inThe octupling case.

Recursive Double-Size Modular Multiplications without Extra Cost for Their Quotients

- Computer Science, MathematicsCT-RSA
- 2009

This paper addresses the computation cost and improves on previous 2l-bit modular multiplication algorithms to return not only the remainder but also the quotient, resulting in an higher performance in the recursive approach, which becomes twice faster in the quadrupling case and four times faster inThe octupling case.

Area and time efficient modular multiplication of large integers

- Computer Science, MathematicsProceedings IEEE International Conference on Application-Specific Systems, Architectures, and Processors. ASAP 2003
- 2003

By use of a small amount of precomputing the loop of A2 can be modified such that the effort within the loop is minimised, which leads to the algorithm A3 and it is verified.

Montgomery Modular Multiplication inResidue

- Computer Science, Mathematics
- 2000

A new RNS modular multiplication for very large operands is presented, based on Montgomery's method adapted to residue arithmetic, which achieves an effect corresponding to a redundant high-radix implementation by choosing the moduli of the RNS system reasonably large.

Low-Weight Polynomial Form Integers for Efficient Modular Multiplication

- MathematicsIEEE Transactions on Computers
- 2007

This work modify GMN by removing restriction on the choice of t and restricting the coefficients of f(t) to 0 and plusmn1, and shows an efficient modular multiplication method using LWPFI moduli.

## References

SHOWING 1-10 OF 12 REFERENCES

A carry-free algorithm for finding the greatest common divisor of two integers

- Mathematics, Computer Science
- 1983

Theorems on factorization and primality testing

- Computer Science
- 1974

This paper is concerned with the problem of obtaining theoretical estimates for the number of arithmetical operations required to factorize a large integer n or test it for primality, and uses a multi-tape Turing machine for this purpose.

Fast Probabilistic Algorithms for Verification of Polynomial Identities

- Computer ScienceJ. ACM
- 1980

Vanous fast probabdlsttc algonthms, with probability of correctness guaranteed a prion, are presented for testing polynomial ldentmes and propemes of systems of polynomials and ancdlary fast algorithms for calculating resultants and Sturm sequences are given.

A method for obtaining digital signatures and public-key cryptosystems

- Computer Science, MathematicsCACM
- 1978

An encryption method is presented with the novel property that publicly revealing an encryption key does not thereby reveal the corresponding decryption key, soriers or other secure means are not needed to transmit keys.

A monte carlo method for factorization

- Computer Science
- 1975

A novel factorization method involving probabilistic ideas is described briefly, and it is suggested that this method should be considered as a viable alternative to traditional factorization methods.

A redundant number system that speeds up modular arithmetic

- 1983

A redundant number system that speeds up modular arithmetic Abstract 801-10-427, Abstracts Amer

- Math. Soc
- 1983

A redundant number system that speeds up modular arithmetic," Abstract 801-10-427

- Abstracts Amer. Math. Soc, v
- 1983

Simmons , " A redundant number system that speeds up modular arithmetic

- 1980

Purdy , " A carry - free algorithm for finding the greatest common divisor of two integers , " Comput

- 1975