scholarly journals Computing canonical heights on elliptic curves in quasi-linear time

2016 ◽  
Vol 19 (A) ◽  
pp. 391-405 ◽  
Author(s):  
J. Steffen Müller ◽  
Michael Stoll
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Linear Time ◽  
Integer Factorization ◽  
Canonical Height ◽  
The Difference

We introduce an algorithm that can be used to compute the canonical height of a point on an elliptic curve over the rationals in quasi-linear time. As in most previous algorithms, we decompose the difference between the canonical and the naive height into an archimedean and a non-archimedean term. Our main contribution is an algorithm for the computation of the non-archimedean term that requires no integer factorization and runs in quasi-linear time.

scholarly journals Points of low height on elliptic surfaces with torsion

2010 ◽  
Vol 13 ◽  
pp. 370-387
Author(s):  
Sonal Jain
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Open Subset ◽  
Function Field ◽  
Rational Point ◽  
Torsion Point ◽  
Integral Points ◽  
Elliptic Surfaces ◽  
Canonical Height

AbstractWe determine the smallest possible canonical height$\hat {h}(P)$for a non-torsion pointPof an elliptic curveEover a function field(t) of discriminant degree 12nwith a 2-torsion point forn=1,2,3, and with a 3-torsion point forn=1,2. For eachm=2,3, we parametrize the set of triples (E,P,T) of an elliptic curveE/with a rational pointPandm-torsion pointTthat satisfy certain integrality conditions by an open subset of2. We recover explicit equations for all elliptic surfaces (E,P,T) attaining each minimum by locating them as curves in our projective models. We also prove that forn=1,2 , these heights are minimal for elliptic curves over a function field of any genus. In each case, the optimal (E,P,T) are characterized by their patterns of integral points.

scholarly journals LANG'S CONJECTURE AND SHARP HEIGHT ESTIMATES FOR THE ELLIPTIC CURVES y2 = x3 + ax

2013 ◽  
Vol 09 (05) ◽  
pp. 1141-1170 ◽  
Author(s):  
PAUL VOUTIER ◽  
MINORU YABUTA
Keyword(s):  
Lower Bound ◽  
Elliptic Curves ◽  
Lower Bounds ◽  
Rational Points ◽  
Upper And Lower Bounds ◽  
Canonical Height ◽  
The Difference ◽  
Lang's Conjecture ◽  
Lang’S Conjecture ◽  
Logarithmic Height

For elliptic curves given by the equation Ea : y2 = x3 + ax, we establish the best-possible version of Lang's conjecture on the lower bound for the canonical height of non-torsion rational points along with best-possible upper and lower bounds for the difference between the canonical and logarithmic height.

PERSPECTIVES OF ELLIPTICAL CRYPTOGRAPHY TO ENSURE THE INTEGRITY AND CONFIDENTIALITY OF INFORMATION

2019 ◽  
Author(s):  
Anna ILYENKO ◽  
Sergii ILYENKO ◽  
Yana MASUR
Keyword(s):  
Information Systems ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Computational Efficiency ◽  
Digital Signature ◽  
Discrete Logarithm ◽  
Algebraic Groups ◽  
Schnorr Signature ◽  
Stability Of Algorithms

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.

scholarly journals Primitive divisors of sequences associated to elliptic curves with complex multiplication

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Matteo Verzobio
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Complex Multiplication ◽  
Dedekind Domain ◽  
Torsion Point ◽  
Integral Ideal ◽  
Primitive Divisors ◽  
Primitive Divisor ◽  
The Ideal

AbstractLet P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

scholarly journals Equidistribution Among Cosets of Elliptic Curve Points in Intervals

2020 ◽  
Vol 14 (1) ◽  
pp. 339-345
Author(s):  
Taechan Kim ◽  
Mehdi Tibouchi
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Character Sums ◽  
Fault Analysis ◽  
Signature Schemes ◽  
Large Subgroup

AbstractIn a recent paper devoted to fault analysis of elliptic curve-based signature schemes, Takahashi et al. (TCHES 2018) described several attacks, one of which assumed an equidistribution property that can be informally stated as follows: given an elliptic curve E over 𝔽q in Weierstrass form and a large subgroup H ⊂ E(𝔽q) generated by G(xG, yG), the points in E(𝔽q) whose x-coordinates are obtained from xG by randomly flipping a fixed, sufficiently long substring of bits (and rejecting cases when the resulting value does not correspond to a point in E(𝔽q)) are close to uniformly distributed among the cosets modulo H. The goal of this note is to formally state, prove and quantify (a variant of) that property, and in particular establish sufficient bounds on the size of the subgroup and on the length of the substring of bits for it to hold. The proof relies on bounds for character sums on elliptic curves established by Kohel and Shparlinski (ANTS–IV).

scholarly journals PARAMETRIZING ELLIPTIC CURVES BY MODULAR UNITS

2015 ◽  
Vol 100 (1) ◽  
pp. 33-41 ◽  
Author(s):  
FRANÇOIS BRUNAULT
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Recent Work ◽  
Short Note ◽  
Torsion Subgroup ◽  
Modular Functions ◽  
Open Questions ◽  
Modular Units

It is well known that every elliptic curve over the rationals admits a parametrization by means of modular functions. In this short note, we show that only finitely many elliptic curves over $\mathbf{Q}$ can be parametrized by modular units. This answers a question raised by W. Zudilin in a recent work on Mahler measures. Further, we give the list of all elliptic curves $E$ of conductor up to 1000 parametrized by modular units supported in the rational torsion subgroup of $E$. Finally, we raise several open questions.

scholarly journals Constructing elliptic curves from Galois representations

2018 ◽  
Vol 154 (10) ◽  
pp. 2045-2054
Author(s):  
Andrew Snowden ◽  
Jacob Tsimerman
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Representations ◽  
Good Reduction ◽  
Arithmetic Surface

Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse $\ell$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification, rational traces of Frobenius elements, and somewhere not potentially good reduction. We prove that any lisse sheaf of rank two possessing these properties comes from an elliptic curve.

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