4 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
Bounds on non-linear errors for variance computation with stochastic rounding *
The main objective of this work is to investigate non-linear errors and
pairwise summation using stochastic rounding (SR) in variance computation
algorithms. We estimate the forward error of computations under SR through two
methods: the first is based on a bound of the variance and
Bienaym{\'e}-Chebyshev inequality, while the second is based on martingales and
Azuma-Hoeffding inequality. The study shows that for pairwise summation, using
SR results in a probabilistic bound of the forward error proportional to
log(n)u rather than the deterministic bound in O(log(n)u) when using the
default rounding mode. We examine two algorithms that compute the variance,
called ''textbook'' and ''two-pass'', which both exhibit non-linear errors.
Using the two methods mentioned above, we show that these algorithms' forward
errors have probabilistic bounds under SR in O(\sqrt nu) instead of nu for
the deterministic bounds. We show that this advantage holds using pairwise
summation for both textbook and two-pass, with probabilistic bounds of the
forward error proportional to log(n)u