Modular multiplication and base extensions in residue number systems (original) (raw)

Montgomery modular multiplication in residue arithmetic

2000

We present a new RNS modular multiplication for very large operands. The algorithm is based on Montgomery's method adapted to residue arithmetic. By choosing the moduli of the RNS system reasonably large, an e ect corresponding to a redundant high-radix implementation is achieved, due to the carry-free nature of residue arithmetic. The actual computation in the multiplication takes place in constant time, where the unit of time is a few simple residue operations. However, it is necessary twice to convert values from one residue system into another, operations which takes O(n) time on O(n) processors, where n is the number of moduli in the RNS systems. Thus these conversions are the bottlenecks of the method, and any future improvements in RNS base conversions, or the use of particular residue systems, can immediately be applied.

An RNS Montgomery modular multiplication algorithm

IEEE Transactions on Computers, 1998

We present a new RNS modular multiplication for very large operands. The algorithm is based on Montgomery's method adapted to mixed radix, and is performed using a Residue Number System. By choosing the moduli of the RNS system reasonably large and implementing the system on a ring of fairly simple processors, an effect corresponding to a redundant high-radix implementation is achieved. The algorithm can be implemented to run in 2(n) time on 2(n) processors, where n is the number of moduli in the RNS system, and the unit of time is a simple residue operation, possibly by table look-up. Two different implementations are proposed, one based on processors attached to a broadcast bus, another on an oriented ring structure.

Novel RNS Parameter Selection for Fast Modular Multiplication

IEEE Transactions on Computers, 2000

The parameter selection of Residue Number Systems (RNS) has a great impact on its computational efficiency. This paper shows that a base extension, the most costly operation in RNS Montgomery multiplication, can be more efficient when the intervals between the RNS moduli are small. We propose a systematic RNS parameter selection procedure and two methods to select RNS moduli that lead to a reduced complexity. Our experimental results confirm the advantages of the selected moduli.

Fast Modular Multiplication Execution in Residue Number System

— In the paper, we propose a new method of modular multiplication computation, based on Residue Number System. We use an approximate method to find the approximate method a residue from division of a multiplication on the given module. We substitute expensive modular operations, by fast bit right shift operations and taking low bits. The carried-out simulation on Kintex7 XC7K70T board showed that the offered method allows to win in time on average for 75%, and in the area-on average for 80% relatively to modified method from work [1] that makes it more applicable for the hardware implementation of the cryptography primitives constructed over a simple finite field.

Four Moduli RNS Bases for Efficient Design of Modular Multiplication

2011

Residue Number System provides parallel and fast arithmetic operation by replacing large number computation with small moduli without carry propagation between moduli. RNS can be applied in application like public key cryptography in order to achieve more speed and less power consumption. Modular Multiplication is the main operation in this application. Selecting RNS moduli sets (bases) is the most important part in modular multiplication. In this work RNS bases in order to design efficient modular multiplication is presented. The proposed RNS bases in first basis employs the basis and multiplicative inverses with small hamming weight based on the work reported in literature and in second basis, well formed arithmetic unit RNS basis with efficient forward and reverse converter are employed. The proposed RNS bases are suitable for public key cryptography algorithm especially for Elliptic Curve Cryptography (ECC). The results show that combination of these RNS basis has achieved noticeable improvement in hardware complexity and also less time delay.

RNS modular multiplication through reduced base extensions

2014 IEEE 25th International Conference on Application-Specific Systems, Architectures and Processors, 2014

The paper describes a new RNS (residue number system) modular multiplication algorithm, for finite field arithmetic over FP , based on a reduced number of moduli in base extensions with only 3n/2 moduli instead of 2n for standard ones. Our algorithm reduces both the number of elementary modular multiplications (EMMs) and the number of stored precomputations for large asymmetric cryptographic applications such as elliptic curve cryptography or Diffie-Hellman (DH) cryptosystem. It leads to faster operations and smaller circuits.

Novel high-radix Residue Number System multipliers and adders

Radix- modulo multipliers and adders are introduced in this paper. The proposed architectures are shown to require several times less area than previously reported architectures, for particular moduli of operation. The proposed architectures are preferable in an area-time sense for several cases. The complexity reduction is achieved by extending the carry-ignore property of modulo operations to radices higher than 2, but not powers of 2. Detailed hardware complexity models are offered. RNS systems are particularly efficient for executing algorithms which contain a significant amount of multiply-accumulate operations (such as DSP algorithms) even when the unavoidable forward and inverse conversion overhead is considered. Bases of the form

Efficient Implementation of RNS Montgomery Multiplication Using Balanced RNS Bases

Point multiplication is the most important part of elliptic curve cryptography which consumes remarkable time of implementation. Therefore efficiency enhancement of entire system is depending on efficiency of this part. Increasing the efficiency of the modular multiplication improve overall performance of the cryptographic system as it frequency used in some application such as Elliptic Curve Cryptography. By applying Residue Number System (RNS) to Montgomery multiplication as a method for modular multiplication, delay of modular multiplication will be reduced. Appropriate RNS moduli sets replace time consuming operation of multiplication by smaller operations. In this paper two balanced moduli set with proper dynamic range is presented and the efficiency of conversion from RNS to RNS which is the most time consuming part of the Montgomery modular multiplication will be increased.

Improved Sum of Residues Modular Multiplication Algorithm

Cryptography, 2019

Modular reduction of large values is a core operation in most common public-key cryptosystems that involves intensive computations in finite fields. Within such schemes, efficiency is a critical issue for the effectiveness of practical implementation of modular reduction. Recently, Residue Number Systems have drawn attention in cryptography application as they provide a good means for extreme long integer arithmetic and their carry-free operations make parallel implementation feasible. In this paper, we present an algorithm to calculate the precise value of “ X mod p ” directly in the RNS representation of an integer. The pipe-lined, non-pipe-lined, and parallel hardware architectures are proposed and implemented on XILINX FPGAs.