Modular redundant number systems
With the increased use of public key cryptography, faster modular multiplication has become an important cryptographic issue. Almost all public key cryptography, including most elliptic curve systems, use modular multiplication. Modular multiplication, particularly for the large public key modulii, is very slow. Increasing the speed of modular multiplication is almost synonymous with increasing the speed of public key cryptography. There are two parts to modular multiplication: multiplication and modular reduction. Though there are fast methods for multiplying and fast methods for doing modular reduction, they do not mix well. Most fast techniques require integers to be in a special form. These special forms are not related and converting from one form to another is more costly than using the standard techniques. To this date it has been better to use the fast modular reduction technique coupled with standard multiplication. Standard modular reduction is much more costly than standard multiplication. Fast modular reduction (Montgomery`s method) reduces the reduction cost to approximately that of a standard multiply. Of the fast multiplication techniques, the redundant number system technique (RNS) is one of the most popular. It is simple, converting a large convolution (multiply) into many smaller independent ones. Not only do redundant number systems increase speed, but the independent parts allow for parallelization. RNS form implies working modulo another constant. Depending on the relationship between these two constants; reduction OR division may be possible, but not both. This paper describes a new technique using ideas from both Montgomery`s method and RNS. It avoids the formula problem and allows fast reduction and multiplication. Since RNS form is used throughout, it also allows the entire process to be parallelized.
- Research Organization:
- Sandia National Laboratories (SNL), Albuquerque, NM, and Livermore, CA (United States)
- Sponsoring Organization:
- USDOE, Washington, DC (United States)
- DOE Contract Number:
- AC04-94AL85000
- OSTI ID:
- 645510
- Report Number(s):
- SAND-97-2732C; CONF-980536-; ON: DE98000847; TRN: AHC2DT03%%51
- Resource Relation:
- Conference: EUROCRYPT `98, Helsinki (Finland), 31 May - 4 Jun 1998; Other Information: PBD: 31 May 1998
- Country of Publication:
- United States
- Language:
- English
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