6 research outputs found
Secure and Efficient RNS Approach for Elliptic Curve Cryptography
Scalar multiplication, the main operation in elliptic
curve cryptographic protocols, is vulnerable to side-channel
(SCA) and fault injection (FA) attacks. An efficient countermeasure
for scalar multiplication can be provided by using alternative
number systems like the Residue Number System (RNS). In RNS,
a number is represented as a set of smaller numbers, where each
one is the result of the modular reduction with a given moduli
basis. Under certain requirements, a number can be uniquely
transformed from the integers to the RNS domain (and vice
versa) and all arithmetic operations can be performed in RNS.
This representation provides an inherent SCA and FA resistance
to many attacks and can be further enhanced by RNS arithmetic
manipulation or more traditional algorithmic countermeasures.
In this paper, extending our previous work, we explore the
potentials of RNS as an SCA and FA countermeasure and provide
an description of RNS based SCA and FA resistance means. We
propose a secure and efficient Montgomery Power Ladder based
scalar multiplication algorithm on RNS and discuss its SCAFA
resistance. The proposed algorithm is implemented on an
ARM Cortex A7 processor and its SCA-FA resistance is evaluated
by collecting preliminary leakage trace results that validate our
initial assumptions
An algorithmic and architectural study on Montgomery exponentiation in RNS
The modular exponentiation on large numbers is computationally intensive. An effective way for performing this operation consists in using Montgomery exponentiation in the Residue Number System (RNS). This paper presents an algorithmic and architectural study of such exponentiation approach. From the algorithmic point of view, new and state-of-the-art opportunities that come from the reorganization of operations and precomputations are considered. From the architectural perspective, the design opportunities offered by well-known computer arithmetic techniques are studied, with the aim of developing an efficient arithmetic cell architecture. Furthermore, since the use of efficient RNS bases with a low Hamming weight are being considered with ever more interest, four additional cell architectures specifically tailored to these bases are developed and the tradeoff between benefits and drawbacks is carefully explored. An overall comparison among all the considered algorithmic approaches and cell architectures is presented, with the aim of providing the reader with an extensive overview of the Montgomery exponentiation opportunities in RNS
Montgomery Arithmetic from a Software Perspective
This chapter describes Peter L. Montgomery\u27s modular multiplication method and the various improvements to reduce the latency for software implementations on devices which have access to many computational units
Selected RNS bases for modular multiplication
The selection of the elements of the bases in an RNS modular multiplication method is crucial and has a great impact in the overall performance.This work proposes specific sets of optimal RNS moduli with elements of Hamming weight three whose inverses used in the MRS reconstruction have very small Hamming weight. This property is exploited in RNS bases conversions, to completely remove and replace the products by few additions/subtractions and shifts, reducing the time complexity of modular multiplication.These bases are specially crafted to computation with operands of sizes 256 or more and are suitable for cryptographic applications such as the ECC protocols