40 research outputs found
Rank zero elliptic curves induced by rational Diophantine triples
Rational Diophantine triples, i.e. rationals a,b,c with the property that
ab+1, ac+1, bc+1 are perfect squares, are often used in construction of
elliptic curves with high rank. In this paper, we consider the opposite problem
and ask how small can be the rank of elliptic curves induced by rational
Diophantine triples. It is easy to find rational Diophantine triples with
elements with mixed signs which induce elliptic curves with rank 0. However,
the problem of finding such examples of rational Diophantine triples with
positive elements is much more challenging, and we will provide the first such
known example.Comment: 7 page
Strong Eulerian triples
We prove that there exist infinitely many rationals a, b and c with the
property that a^2-1, b^2-1, c^2-1, ab-1, ac-1 and bc-1 are all perfect squares.
This provides a solution to a variant of the problem studied by Diophantus and
Euler.Comment: 8 page
Rank zero elliptic curves induced by rational Diophantine triples
Rational Diophantine triples, i.e. rationals a, b, c with the property that ab + 1, ac + 1, bc + 1 are perfect squares, are often used in the construction of elliptic curves with high rank. In this paper, we consider the opposite problem and ask how small can be the rank of elliptic curves induced by rational Diophantine triples. It is easy to find rational Diophantine triples with elements with mixed signs which induce elliptic curves with rank 0. However, the problem of finding such examples of rational Diophantine triples with positive elements is much more challenging, and we will provide the first such known example
On the post-quantum future of Elliptic Curve Cryptography
This thesis is a literature study on current published quantum-resistant isogeny-based key exchange protocols.
Here we cover the topic from foundations. Chapters 1 and 2 discuss classical computation models, algorithm complexity, and how these concepts support the security of modern elliptic curve cryptography methods, such as ECDH and ECDSA.
Next, in Chapters 3 to 5, we present quantum computation models, and how Shor's algorithm on quantum computers presents a threat to the future security of classical asymmetric cryptography. We explore the foundations of isogeny-based cryptography, and two key exchange protocols of this kind: SIDH and CSIDH.
Appendices A and B are provided for readers wanting more in-depth background explanations on the algebraic geometry of elliptic curves, and quantum mechanics respectively
Analogues of Velu\u27s Formulas for Isogenies on Alternate Models of Elliptic Curves
Isogenies are the morphisms between elliptic curves, and are accordingly a topic of interest in the subject. As such, they have been well-studied, and have been used in several cryptographic applications. Veluās formulas show how to explicitly evaluate an isogeny, given a specification of the kernel as a list of points. However, Veluās formulas only work for elliptic curves specified by a Weierstrass equation. This paper presents formulas similar to Veluās that can be used to evaluate isogenies on Edwards curves and Huff curves, which are normal forms of elliptic curves that provide an alternative to the traditional Weierstrass form. Our formulas are not simply compositions of Veluās formulas with mappings to and from Weierstrass form. Our alternate derivation yields efficient formulas for isogenies with lower algebraic complexity than such compositions. In fact, these formulas have lower algebraic complexity than Veluās formulas on Weierstrass curves