43 research outputs found
A Note on Cyclic Codes from APN Functions
Cyclic codes, as linear block error-correcting codes in coding theory, play a
vital role and have wide applications. Ding in \cite{D} constructed a number of
classes of cyclic codes from almost perfect nonlinear (APN) functions and
planar functions over finite fields and presented ten open problems on cyclic
codes from highly nonlinear functions. In this paper, we consider two open
problems involving the inverse APN functions and the Dobbertin
APN function . From the calculation of
linear spans and the minimal polynomials of two sequences generated by these
two classes of APN functions, the dimensions of the corresponding cyclic codes
are determined and lower bounds on the minimum weight of these cyclic codes are
presented. Actually, we present a framework for the minimal polynomial and
linear span of the sequence defined by ,
where is a primitive element in . These techniques can also be
applied into other open problems in \cite{D}
The Weight Distributions of Cyclic Codes and Elliptic Curves
Cyclic codes with two zeros and their dual codes as a practically and
theoretically interesting class of linear codes, have been studied for many
years. However, the weight distributions of cyclic codes are difficult to
determine. From elliptic curves, this paper determines the weight distributions
of dual codes of cyclic codes with two zeros for a few more cases
A New Lever Function with Adequate Indeterminacy
The key transform of the REESSE1+ asymmetrical cryptosystem is Ci = (Ai * W ^
l(i)) ^ d (% M) with l(i) in Omega = {5, 7, ..., 2n + 3} for i = 1, ..., n,
where l(i) is called a lever function. In this paper, the authors give a
simplified key transform Ci = Ai * W ^ l(i) (% M) with a new lever function
l(i) from {1, ..., n} to Omega = {+/-5, +/-6, ..., +/-(n + 4)}, where "+/-"
means the selection of the "+" or "-" sign. Discuss the necessity of the new
l(i), namely that a simplified private key is insecure if the new l(i) is a
constant but not one-to-one function. Further, expound the sufficiency of the
new l(i) from four aspects: (1) indeterminacy of the new l(i), (2) insufficient
conditions for neutralizing the powers of W and W ^-1 even if Omega = {5, 6,
..., n + 4}, (3) verification by examples, and (4) running times of continued
fraction attack and W-parameter intersection attack which are the two most
efficient algorithms of the probabilistic polytime attacks so far. Last, the
authors detail the relation between a lever function and a random oracle.Comment: 13 page
Implementing 4-Dimensional GLV Method on GLS Elliptic Curves with j-Invariant 0
The Gallant-Lambert-Vanstone (GLV) method is a very efficient technique for accelerating point multiplication on elliptic curves with efficiently computable endomorphisms. Galbraith, Lin and Scott (J. Cryptol. 24(3), 446-469 (2011)) showed that point multiplication exploiting the 2-dimensional GLV method on a large class of curves over GF(p^2) was faster than the standard method on general elliptic curves over GF(p), and left as an open problem to study the case of 4-dimensional GLV on special curves (e.g., j(E) = 0) over GF(p^2). We study the above problem in this paper. We show how to get the 4-dimensional GLV decomposition with proper decomposed coefficients, and thus reduce the number of doublings for point multiplication on these curves to only a quarter. The resulting implementation shows that the 4-dimensional GLV method on a GLS curve runs in about 0.78 the time of the 2-dimensional GLV method on the same curve and in between 0.78-0.87 the time of the 2-dimensional GLV method using the standard method over GF(p). In particular, our implementation reduces by up to 27% the time of the previously fastest implementation of point multiplication on x86-64 processors due to Longa and Gebotys (CHES2010)
Efficient Comb Elliptic Curve Multiplication Methods Resistant to Power Analysis
Elliptic Curve Cryptography (ECC) has found wide applications in
smart cards and embedded systems. Point multiplication plays a
critical role in ECC. Many efficient point multiplication methods
have been proposed. One of them is the comb method which
is much more efficient than other methods if precomputation points
are calculated in advance or elsewhere. Unfortunately, Many
efficient point multiplication methods including the comb method are
vulnerable to power-analysis attacks. Various algorithms to make
elliptic curve point multiplication secure to power-analysis attacks
have been proposed recently, such as the double-and-add-always
method, Möller\u27s window method, Okeya
et al.\u27s odd-only window method, and Hedabou et al.\u27s
comb method. In this paper, we first present a novel comb
recoding algorithm which converts an integer to a sequence of
signed, odd-only comb bit-columns. Using this recoding algorithm, we
then present several comb methods, both Simple Power Analysis
(SPA)-nonresistant and SPA-resistant, for point multiplication.
These comb methods are more efficient than the original
SPA-nonresistant comb method and Hedabou et al.\u27s SPA-resistant comb
method. Our comb methods inherit the advantage of a comb method,
running much faster than Möller\u27s window method and Okeya et
al.\u27s odd-only window method, as well as other window methods such
as the efficient signed -ary window method, if only the
evaluation phase is taken into account. Combined with randomization
projective coordinates or other randomization techniques and certain
precautions in selecting elliptic curves and parameters, our
SPA-resistant comb methods are resistant to all power-analysis
attacks
On properties of the Mullineux map with an application to Schur modules
this paper we study a third description of M based on the operator J on the set of p-regular partitions defined in [13