Jacek Pomykała

Rang des courbes elliptiques et sommes d'exponentielles

Lety2=x3+a(t)x+b(t) be the equation of an elliptic curve over ℚ, wherea(t) andb(t) are polynomials overℤ. We give an upper bound on average of the rank of such curves whent varies overℤ under the assumption that theL-function attached to these curves satisfies classical assumptions. The proof is based on estimates of exponential sums, over varieties, which are treated by Delignes theorem.

Multi-threshold signature

The paper presents a novel hybrid spectrum sensing method used in cognitive radio and presents a hybrid detector (HD) which improves the sensing performance. The proposed HD takes advantage of the energy detection (ED) principle and a method based on Covariance Absolute Value (CAV), as well as on Cyclic Autocorrelation Function (CAF). The paper shows the limitations of using ED, resulting from the uncertainty of spectral density of noise power estimation, known as the SNR wall. The paper describes a system model and presents simulation results for the OFDM signal of a WiMAX-based communications system. The simulation results refer to an ideal environment with well-known parameters, and to an environment with uncertain spectral density of noise power estimation. Keywords—Covariance Absolute Value, Cyclic Autocorrelation Function, hybrid detector, noise uncertainty, OFDM, SNR wall, WiMAX.

On Deterministic Reduction of Factoring Integers to Computing the Exponents of Elements in Modular Group

In the paper we prove that all but at most x/A(x) positive integers n ≤ x can be completely factored in deterministic polynomial time C(x), querying the prime decomposition exponent oracle at most D(x) times. The functions A(x), C(x) and D(x) have the polynomial growth (of log x) at infinity.