Related papers: On Using Expansions to the Base of -2
Binary field extensions are fundamental to many applications, such as multivariate public key cryptography, code-based cryptography, and error-correcting codes. Their implementation requires a foundation in number theory and algebraic…
Genus 2 curves have been an object of much mathematical interest since eighteenth century and continued interest to date. They have become an important tool in many algorithms in cryptographic applications, such as factoring large numbers,…
Elliptic curve multiplications can be improved by replacing the standard ladder algorithm's base 2 representation of the scalar multiplicand, with mixed-base representations with power-of-2 bases, processing the n bits of the current digit…
We present e cient algorithms for computing isogenies between hyperelliptic curves, leveraging higher genus curves to enhance cryptographic protocols in the post-quantum context. Our algorithms reduce the computational complexity of isogeny…
The main purpose of this paper is to give an overview over the theory of abelian varieties, with main focus on Jacobian varieties of curves reaching from well-known results till to latest developments and their usage in cryptography. In the…
We give a new algorithm for constructing Picard curves over a finite field with a given endomorphism ring. This has important applications in cryptography since curves of genus 3 allow for smaller key sizes than elliptic curves. For a…
We give a general framework for uniform, constant-time one-and two-dimensional scalar multiplication algorithms for elliptic curves and Jacobians of genus 2 curves that operate by projecting to the x-line or Kummer surface, where we can…
Elliptic curves are planar curves which can be used to define an abelian group. The efficient computation of discrete logarithms over this group is a longstanding problem relevant to cryptography. It may be possible to efficiently compute…
This paper presents explicit constructions of bases for Riemann-Roch spaces associated with arbitrary divisors on elliptic curves. In the context of algebraic geometry codes, the knowledge of an explicit basis for arbitrary divisors is…
Pairing based cryptography is in a dangerous position following the breakthroughs on discrete logarithms computations in finite fields of small characteristic. Remaining instances are built over finite fields of large characteristic and…
In the study of hyperelliptic curve cryptography, presentations of semi-reduced divisors on a hyperelliptic curve play important roles. In this note, we give an interpretation for such presentations from view points of Gr\"obner bases. As…
In this paper we will give various examples of exponentially distorted subgroups in linear groups, including some new example of subgroups of $SL_n(\mathbb{Z}[x])$ for $n \ge 3$, and show how they can be used to construct symmetric-key…
In this article we give the details of an effective point counting algorithm for genus two curves over finite fields of characteristic three. The algorithm has an application in the context of curve based cryptography. One distinguished…
Expander decompositions of graphs have significantly advanced the understanding of many classical graph problems and led to numerous fundamental theoretical results. However, their adoption in practice has been hindered due to their…
We present normal forms for elliptic curves over a field of characteristic $2$ analogous to Edwards normal form, and determine bases of addition laws, which provide strikingly simple expressions for the group law. We deduce efficient…
Most common public key cryptosystems and public key exchange protocols presently in use, such as the RSA algorithm, Diffie-Hellman, and elliptic curve methods are number theory based and hence depend on the structure of abelian groups. The…
Shor's quantum algorithm for discrete logarithms applied to elliptic curve groups forms the basis of a "quantum attack" of elliptic curve cryptosystems. To implement this algorithm on a quantum computer requires the efficient implementation…
We present a construction of expander graphs obtained from Cayley graphs of narrow ray class groups, whose eigenvalue bounds follow from the Generalized Riemann Hypothesis. Our result implies that the Cayley graph of (Z/qZ)* with respect to…
In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. In this paper, we slightly modify this notion to obtain the so-called irreducible-expansion complexity which is more suitable for…
Edge expansion is a parameter indicating how well-connected a graph is. It is useful for designing robust networks, analysing random walks or information flow through a network and is an important notion in theoretical computer science.…