Dimitar Jetchev

How to Fake Auxiliary Input

Consider a joint distribution (X,A) on a set \({\mathcal X}\times\{0, 1\}^\ell\). We show that for any family \({\mathcal F}\) of distinguishers \(f : {\cal X} \times \{0, 1\}^\ell \rightarrow\{0,1\}\), there exists a simulator \(h : {\mathcal X} \rightarrow \{0,1\}^\ell\) such that

Global divisibility of Heegner points and Tamagawa numbers

We improve Kolyvagin’s upper bound on the order of the p -primary part of the Shafarevich–Tate group of an elliptic curve of rank one over a quadratic imaginary field. In many cases, our bound is precisely that predicted by the Birch and Swinnerton-Dyer conjectural formula.

Bits Security of the Elliptic Curve Diffie---Hellman Secret Keys

We show that the least significant bits (LSB) of the elliptic curve Diffie---Hellman secret keys are hardcore. More precisely, we prove that if one can efficiently predict the LSB with non-negligible advantage on a polynomial fraction of all the curves defined over a given finite field

On the bits of elliptic curve Diffie-Hellman Keys

\mathbb{F}_p

Equidistribution of Heegner points and ternary quadratic forms

, then with polynomial factor overhead, one can compute the entire Diffie---Hellman secret on a polynomial fraction of all the curves over the same finite field. Our approach is based on random self-reducibility (assuming GRH) of the Diffie---Hellman problem among elliptic curves of the same order. As a part of the argument, we prove a refinement of H. W. Lenstras lower bounds on the sizes of the isogeny classes of elliptic curves, which may be of independent interest.

Collision bounds for the additive Pollard rho algorithm for solving discrete logarithms

We study the security of elliptic curve Diffie-Hellman secret keys in the presence of oracles that provide partial information on the value of the key. Unlike the corresponding problem for finite fields, little is known about this problem, and in the case of elliptic curves the difficulty of representing large point multiplications in an algebraic manner leads to new obstacles that are not present in the case of finite fields. To circumvent this obstruction, we introduce a small multiplier version of the hidden number problem, and we use its properties to analyze the security of certain Diffie-Hellman bits. We suggest new character sum conjectures that guarantee the uniqueness of solutions to the hidden number problem, and provide some evidence in support of the conjectures by showing that they hold on average in certain cases. We also present a Grobner basis algorithm for solving the hidden number problem and recovering the Diffie-Hellman secret key when the elliptic curve is defined over a constant degree extension field and the oracle is a coordinate function in the polynomial basis.

Liftings of reduction maps for quaternion algebras

We prove new equidistribution results for Galois orbits of Heegner points with respect to single reduction maps at inert primes. The arguments are based on two different techniques: primitive representations of integers by quadratic forms and distribution relations for Heegner points. Our results generalize an equidistribution result with respect to a single reduction map established by Cornut and Vatsal in the sense that we allow both the fundamental discriminant and the conductor to grow. Moreover, for fixed fundamental discriminant and variable conductor, we deduce an effective surjectivity theorem for the reduction map from Heegner points to supersingular points at a fixed inert prime. Our results are applicable to the setting considered by Kolyvagin in the construction of the Heegner points Euler system.

Computing the Cassels Pairing on Kolyvagin Classes in the Shafarevich-Tate Group

Abstract. We prove collision bounds for the Pollard rho algorithm to solve the discrete logarithm problem in a general cyclic group 𝐆

Isogeny graphs of ordinary abelian varieties

\mathbf {G}

The Birch and Swinnerton–Dyer formula for elliptic curves of analytic rank one

. Unlike the setting studied by Kim et al., we consider additive walks: the setting used in practice to solve the elliptic curve discrete logarithm problem. Our bounds differ from the birthday bound 𝒪(|𝐆|)