Scalar multiplication | EPFL Graph Search

In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector—without changing its direction. The term "scalar" itself derives from this usage: a scalar is that which scales vectors. Scalar multiplication is the multiplication of a vector by a scalar (where the product is a vector), and is to be distinguished from inner product of two vectors (where the product is a scalar). Definition In general, if K is a field and V is a vector space over K, then scalar multiplication is a function from K × V to V. The result of applying this function to k in K and v in V is denoted kv. Properties Scalar multiplication obeys the following rules (vector in boldface):

Additivity in the scalar: (c + d)v = cv + dv;
Additivity in

Computational Aspects of Jacobians of Hyperelliptic Curves

Alina Dudeanu

Nowadays, one area of research in cryptanalysis is solving the Discrete Logarithm Problem (DLP) in finite groups whose group representation is not yet exploited. For such groups, the best one can do is using a generic method to attack the DLP, the fastest of which remains the Pollard rho algorithm with

r

-adding walks. For the first time, we rigorously analyze the Pollard rho method with

r

-adding walks and prove a complexity bound that differs from the birthday bound observed in practice by a relatively small factor. There exist a multitude of open questions in genus

2

cryptography. In this case, the DLP is defined in large prime order subgroups of rational points that are situated on the Jacobian of a genus~

2

curve defined over a large characteristic finite field. We focus on one main topic, namely we present a new efficient algorithm for computing cyclic isogenies between Jacobians. Comparing to previous work that computes non cyclic isogenies in genus~

2

, we need to restrict to certain cases of polarized abelian varieties with specific complex multiplication and real multiplication. The algorithm has multiple applications related to the structure of the isogeny graph in genus~

2

, including random self-reducibility of DLP. It helps support the widespread intuition of choosing \emph{any} curve in a class of curves that satisfy certain public and well studied security parameters. Another topic of interest is generating hyperelliptic curves for cryptographic applications via the CM method that is based on the numerical estimation of the rational Igusa class polynomials. A recent development relates the denominators of the Igusa class polynomials to counting ideal classes in non maximal real quadratic orders whose norm is not prime to the conductor. Besides counting, our new algorithm provides precise representations of such ideal classes for all real quadratic fields and is part of an implementation in Magma of the recent theoretic work in the literature on the topic of denominators.

EPFL2016

Low-Latency Elliptic Curve Scalar Multiplication

Joppe Willem Bos

This paper presents a low-latency algorithm designed for parallel computer architectures to compute the scalar multiplication of elliptic curve points based on approaches from cryptographic side-channel analysis. A graphics processing unit implementation using a standardized elliptic curve over a 224-bit prime field, complying with the new 112-bit security level, computes the scalar multiplication in 1.9 ms on the NVIDIA GTX 500 architecture family. The presented methods and implementation considerations can be applied to any parallel 32-bit architecture.

Springer Verlag2012

Nilpotent centralizers and Springer isomorphisms

Donna Testerman

Let GG be a semisimple algebraic group over a field KK whose characteristic is very good for GG, and let σσ be any GG-equivariant isomorphism from the nilpotent variety to the unipotent variety; the map σσ is known as a Springer isomorphism. Let y∈G(K)y∈G(K), let Y∈Lie(G)(K)Y∈Lie(G)(K), and write Cy=CG(y)Cy=CG(y) and CY=CG(Y)CY=CG(Y) for the centralizers. We show that the center of CyCy and the center of CYCY are smooth group schemes over KK. The existence of a Springer isomorphism is used to treat the crucial cases where yy is unipotent and where YY is nilpotent. Now suppose GG to be quasisplit, and write CC for the centralizer of a rational regular nilpotent element. We obtain a description of the normalizer NG(C)NG(C) of CC, and we show that the automorphism of Lie(C)Lie(C) determined by the differential of σσ at zero is a scalar multiple of the identity; these results verify observations of J.-P. Serre.

2009