We present a polynomial-time algorithm that provably recovers the signer's secret DSA key when a few consecutive bits of the random nonces k (used at each signature generation) are known for a number of DSA signatures at most linear in log q (q denoting as usual the small prime of DSA), under a reasonable assumption on the hash function used in DSA. For most significant or least significant bits, the number of required bits is about log 1/2 q, but can be decreased to log log q with a running time q O(1/log log q) subexponential in log q, and even further to two in polynomial time if one assumes access to ideal lattice basis reduction, namely an oracle for the lattice closest vector problem for the infinity norm. For arbitrary consecutive bits, the attack requires twice as many bits. All previously known results were only heuristic, including those of Howgrave-Graham and Smart who recently introduced that topic. Our attack is based on a connection with the hidden number problem (HNP) introduced at Crypto '96 by Boneh and Venkatesan in order to study the bit-security of the Diffie-Hellman key exchange. The HNP consists, given a prime number q, of recovering a number α ∈ F q such that for many known random t ∈ F q a certain approximation of tα is known. To handle the DSA case, we extend Boneh and Venkatesan's results on the HNP to the case where t has not necessarily perfectly uniform distribution, and establish uniformity statements on the DSA signatures, using exponential sum techniques. The efficiency of our attack has been validated experimentally, and illustrates once again the fact that one should be very cautious with the pseudo-random generation of the nonce within DSA.

No abstract

The intention of the paper is to move a step towards a classification of network topologies that exhibit periodic quantum dynamics. We show that the evolution of a quantum system, whose hamiltonian is identical to the adjacency matrix of a circulant graph, is periodic if and only if all eigenvalues of the graph are integers (that is, the graph is integral ). Motivated by this observation, we focus on relevant properties of integral circulant graphs. Specifically, we bound the number of vertices of integral circulant graphs in terms of their degree, characterize bipartiteness and give exact bounds for their diameter. Additionally, we prove that circulant graphs with odd order do not allow perfect state transfer.

We show that for a prime p the smallest a with a p−1 ≡ 1 (mod p 2 ) does not exceed (log p) 463/252+o(1) which improves the previous bound O((log p) 2 ) obtained by H. W. Lenstra in 1979. We also show that for almost all primes p the bound can be improved as (log p) 5/3+o(1) .

We obtain asymptotic formulae for the number of primes p x for which the reduction modulo p of the elliptic curvesatisfies certain "natural" properties, on average over integers a and b such that |a| A and |b| B, where A and B are small relative to x. More precisely, we investigate behavior with respect to the Sato-Tate conjecture, cyclicity, and divisibility of the number of points by a fixed integer m.