Let E be an elliptic curve over Q and let p ≥ 5 be a prime of good supersingular reduction for E. Let K be an imaginary quadratic field satisfying a modified “Heegner hypothesis” in which p splits, write K∞ for the anticyclotomic Zp-extension of K and let Λ denote the Iwasawa algebra of K∞/K. By extending to the supersingular case the Λ-adic Kolyvagin method originally developed by Bertolini in...

Without the hypothesis “ordinary” the answer can be no, because for a supersingular elliptic curve one can have A(n) = 0 on entire arithmetic progressions of n. On the other hand, all the A(n) in the supersingular case are divisible, as algebraic integers, by qn/2, so the nonzero A(n) must have |A(n)| ≥ qn/2. If instead E/Fq is ordinary, then all the A(n) are nonzero because they are all prime ...

‎By the Mordell‎- ‎Weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎This paper studies the rank of the family Epq:y2=x3-pqx of elliptic curves‎, ‎where p and q are distinct primes‎. ‎We give infinite families of elliptic curves of the form y2=x3-pqx with rank two‎, ‎three and four‎, ‎assuming a conjecture of Schinzel ...

‎‎in the category of mordell curves (e_d:y^2=x^3+d) with nontrivial torsion groups we find curves of the generic rank two as quadratic twists of (e_1), ‎and of the generic rank at least two and at least three as cubic twists of (e_1). ‎previous work‎, ‎in the category of mordell curves with trivial torsion groups‎, ‎has found infinitely many elliptic curves with rank at least seven as sextic tw...

We give a brief survey of elliptic curve isogenies and the computational problems relevant for supersingular isogeny crypto. Supersingular isogeny cryptography is attracting attention due to the fact that there are no quantum attacks known against it that are significantly faster than classical attacks. However, the underlying computational problems have not been sufficiently studied by quantum...

There is a recent trend in cryptography to construct protocols based on the hardness of computing isogenies between supersingular elliptic curves. Two prominent examples are Jao-De Feo’s key exchange protocol and the resulting encryption scheme by De Feo-Jao-Plût. One particularity of the isogeny problems underlying these protocols is that some additional information is given as input, namely t...

The parameter k always stands for a non-negative integer or half an integer throughout the paper. This differential equation originates in the work [1] where in some cases (k ≡ 0, 4 mod 6) solutions which are modular on SL2(Z) were found and studied in connection with liftings of supersingular j-invariants of elliptic curves. The purpose of this paper is to give an explicit description of (conj...

This paper presents an efficient architecture for computing cryptographic ηT pairing for providing 128-bit security. A cryptoprocessor is proposed for Miller’s Algorithm with a new 1223-bit Karatsuba multiplier that exploits parallelism. To the best of our knowledge this is the first hardware implementation of 128-bit secure ηT pairing on supersingular elliptic curves over characteristic two fi...

Nowadays the generation of cryptosystems requires two main aspects. First the security, and then the size of the keys involved in the construction and comunication process. About the former one needs a difficult mathematical assumption which ensures your system will not be broken unless a well known difficult problem is solved. In this context one of the most famous assumption underlying a wide...