302 research outputs found
A Novel Method of Encryption using Modified RSA Algorithm and Chinese Remainder Theorem
Security can only be as strong as the weakest link. In this world of cryptography, it is now well established, that the weakest link lies in the implementation of cryptographic algorithms. This project deals with RSA algorithm implementation with and without Chinese Remainder Theorem and also using Variable Radix number System. In practice, RSA public exponents are chosen to be small which makes encryption and signature verification reasonably fast. Private exponents however should never be small for obvious security reasons. This makes decryption slow. One way to speed things up is to split things up, calculate modulo p and modulo q using Chinese Remainder Theorem. For smart cards which usually have limited computing power, this is a very important and useful technique. This project aims at implementing RSA algorithm using Chinese Remainder Theorem as well as to devise a modification using which it would be still harder to decrypt a given encrypted message by employing a Variable radix system in order to encrypt the given message at the first place
Higher dimensional 3-adic CM construction
We find equations for the higher dimensional analogue of the modular curve
X_0(3) using Mumford's algebraic formalism of algebraic theta functions. As a
consequence, we derive a method for the construction of genus 2 hyperelliptic
curves over small degree number fields whose Jacobian has complex
multiplication and good ordinary reduction at the prime 3. We prove the
existence of a quasi-quadratic time algorithm for computing a canonical lift in
characteristic 3 based on these equations, with a detailed description of our
method in genus 1 and 2.Comment: 23 pages; major revie
Elliptical Curve Digital Signatures Algorithm
Elliptical digital signatures algorithm provides security services for resource constrained embedded devices. The ECDSA level security can be enhanced by several parameters as parameter key size and the security level of ECDSA elementary modules such as hash function, elliptic curve point multiplication on koblitz curve which is used to compute public key and a pseudo-random generator which generates key pair generation. This paper describes novel security approach on authentication schemes as a modification of ECDSA scheme. This paper provides a comprehensive survey of recent developments on elliptic curve digital signatures approaches. The survey of ECDSA involves major issues like security of cryptosystem, RFID-tag authentication, Montgomery multiplication over binary fields, Scaling techniques, Signature generation ,signature verification, point addition and point doubling of the different coordinate system and classification.
DOI: 10.17762/ijritcc2321-8169.150318
Minimal Polynomials of Singular Moduli
Given a properly normalized parametrization of a genus-0 modular curve, the
complex multiplication points map to algebraic numbers called singular moduli.
In the classical case, the maps can be given analytically. However, in the
Shimura curve cases, no such analytical expansion is possible. Fortunately, in
both cases there are known algorithms for algebraically computing the rational
norms of the singular moduli. We demonstrate a method of using these norm
algorithms to algebraically determine the minimal polynomial of the singular
moduli below a discriminant threshold. We then use these minimal polynomials to
compute the algebraic -ratios for the singular moduli.Comment: 9 pages, 2 figure
Curves, Jacobians, and Cryptography
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 first part we provide the necessary mathematical
background on abelian varieties, their torsion points, Honda-Tate theory,
Galois representations, with emphasis on Jacobian varieties and hyperelliptic
Jacobians. In the second part we focus on applications of abelian varieties on
cryptography and treating separately, elliptic curve cryptography, genus 2 and
3 cryptography, including Diffie-Hellman Key Exchange, index calculus in Picard
groups, isogenies of Jacobians via correspondences and applications to discrete
logarithms. Several open problems and new directions are suggested.Comment: 66 page
Explicit CM-theory for level 2-structures on abelian surfaces
For a complex abelian variety with endomorphism ring isomorphic to the
maximal order in a quartic CM-field , the Igusa invariants generate an abelian extension of the reflex field of . In
this paper we give an explicit description of the Galois action of the class
group of this reflex field on . We give a geometric
description which can be expressed by maps between various Siegel modular
varieties. We can explicitly compute this action for ideals of small norm, and
this allows us to improve the CRT method for computing Igusa class polynomials.
Furthermore, we find cycles in isogeny graphs for abelian surfaces, thereby
implying that the `isogeny volcano' algorithm to compute endomorphism rings of
ordinary elliptic curves over finite fields does not have a straightforward
generalization to computing endomorphism rings of abelian surfaces over finite
fields
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