Rational torsion on optimal curves and rank-one quadratic twists

AbstractWhen an elliptic curve E′/Q of square-free conductor N has a rational point of odd prime order l∤N, Dummigan (2005) in [Du] explicitly constructed a rational point of order l on the optimal curve E, isogenous over Q to E′, under some conditions. In this paper, we show that his construction also works unconditionally. And applying it to Heegner points of elliptic curves, we find a family of elliptic curves E′/Q such that a positive proportion of quadratic twists of E′ has (analytic) rank 1. This family includes the infinite family of elliptic curves of the same property in Byeon, Jeon, and Kim (2009) [B-J-K]

ON THE JACOBIAN OF A FAMILY OF HYPERELLIPTIC CURVES

In this paper, we study the algebraic rank and the analytic rank of the Jacobian of hyperelliptic curves y² = x⁵ + m² for integers m. Namely, we first provide a condition on m that gives a bound of the size of Selmer group and then we provide a condition on m that makes L-functions non-vanishing. As a consequence, we construct a Jacobian that satisfies the rank part of the Birch–Swinnerton-Dyer conjecture

Multi-dimensional Packing for HEAAN for Approximate Matrix Arithmetics

HEAAN is a homomorphic encryption (HE) scheme for approximate arithmetics. Its vector packing technique proved its potential in cryptographic applications requiring approximate computations, including data analysis and machine learning. In this paper, we propose MHEAAN - a generalization of HEAAN to the case of a tensor structure of plaintext slots. Our design takes advantage of the HEAAN scheme, that the precision losses during the evaluation are limited by the depth of the circuit, and it exceeds no more than one bit compared to unencrypted approximate arithmetics, such as floating point operations. Due to the multi-dimensional structure of plaintext slots along with rotations in various dimensions, MHEAAN is a more natural choice for applications involving matrices and tensors. We provide a concrete two-dimensional construction and show the efficiency of our scheme on several matrix operations, such as matrix multiplication, matrix transposition, and inverse. As an application, we implement the non-interactive Deep Neural Network (DNN) classification algorithm on encrypted data and encrypted model. Due to our efficient bootstrapping, the implementation can be easily extended to DNN structure with an arbitrary number of hidden layer