Elliptic Curves

In this article, the main problems underlying the current asymmetric crypto algorithms for the formation and verification of electronic-digital signature are considered: problems of factorization of large integers and problems of discrete logarithm. It is noted that for the second problem, it is possible to use algebraic groups of points other than finite fields. The group of points of the elliptical curve, which satisfies all set requirements, looked attractive on this side. Aspects of the application of elliptic curves in cryptography and the possibilities offered by these algebraic groups in terms of computational efficiency and crypto-stability of algorithms were also considered. Information systems using elliptic curves, the keys have a shorter length than the algorithms above the finite fields. Theoretical directions of improvement of procedure of formation and verification of electronic-digital signature with the possibility of ensuring the integrity and confidentiality of information were considered. The proposed method is based on the Schnorr signature algorithm, which allows data to be recovered directly from the signature itself, similarly to RSA-like signature systems, and the amount of recoverable information is variable depending on the information message. As a result, the length of the signature itself, which is equal to the sum of the length of the end field over which the elliptic curve is determined, and the artificial excess redundancy provided to the hidden message was achieved.

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Structured digital multisignature algorithm based on elliptic curves

Journal of Computer Applications ◽

10.3724/sp.j.1087.2009.00158 ◽

2009 ◽

Vol 29 (1) ◽

pp. 158-160

Author(s):

He-gang FU ◽

Ying CHEN

Keyword(s):

Elliptic Curves

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Algorithm for directly computing 7P elliptic curves and its application

Journal of Computer Applications ◽

10.3724/sp.j.1087.2013.01870 ◽

2013 ◽

Vol 33 (7) ◽

pp. 1870-1874

Author(s):

Zhongxi LAI ◽

Zhanjun ZHANG ◽

Dongya TAO

Keyword(s):

Elliptic Curves

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Elliptic Curves. (MN-40), Volume 40

10.2307/j.ctv346st5 ◽

2018 ◽

Author(s):

Anthony W. Knapp

Keyword(s):

Elliptic Curves

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Mazur’s rational torsion result for pointless genus one curves: examples

Research in Number Theory ◽

10.1007/s40993-020-00231-z ◽

2021 ◽

Vol 7 (1) ◽

Author(s):

Arjan Dwarshuis ◽

Majken Roelfszema ◽

Jaap Top

Keyword(s):

Algebraic Geometry ◽

Elliptic Curves ◽

Rational Point ◽

Torsion Points ◽

Arbitrary Genus ◽

Mazur’S Theorem ◽

Genus One

AbstractThis note reformulates Mazur’s result on the possible orders of rational torsion points on elliptic curves over $$\mathbb {Q}$$ Q in a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of ‘pointless’ genus one curves over $$\mathbb {Q}$$ Q corresponding to the torsion orders 7, 8, 9, 10, 12 (and hence, all possibilities) occurring in Mazur’s theorem. In fact three distinct methods are proposed for constructing such examples, each involving different in our opinion quite nice ideas from the arithmetic of elliptic curves or from algebraic geometry.

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