The general number field sieve (GNFS) is asymptotically the fastest known factoring algorithm. One of the most important steps of GNFS is to select a good polynomial pair. A standard way of polynomial selection (being used in factoring RSA challenge numbers) is to select a nonlinear polynomial for algebraic sieving and a linear polynomial for rational sieving. There is another method called a n...

Golay complementary sequences, often referred to as Golay pairs, are characterised by the property that the sum of their aperiodic autocorrelation functions equals to zero, except for the zero shift. Because of this property, Golay complementary sequences can be utilised to construct Hadamard matrices defining sets of orthogonal spreading sequences for DS CDMA systems of the lengths not necessa...

Multicarrier transmission system such as Orthogonal Frequency Division Multiplexing (OFDM) is a promising technique for high bit rate transmission in wireless communication systems. OFDM is a spectrally efficient modulation technique that can achieve high speed data transmission over multipath fading channels without the need for powerful equalization techniques. A major drawback of OFDM is the...

Polynomials in differentiation operators are considered. The Darboux transformations covariance determines non-Abelian entries to form the coefficients of the polynomials. Joint covariance of a pair of such polynomials (Lax pair) as a function of one-field is studied. Methodically, the transforms of the coefficients are equalized to Frechet derivatives (first term of the Taylor series on prolon...

If the yield of a polynomial pair is closely correlated with the coefficients of the polynomial pair, we can select polynomials by checking the coefficients first. This can speed the selection of good polynomials. In this paper, we aim to study the correlation between the polynomial coefficients and the yield of the polynomials. By heuristic analysis and some experiments, we find that the yield...

We present a new method for building pairs of HFE polynomials of high degree, such that the map constructed with such a pair is easy to invert. The inversion is accomplished using a low degree polynomial of Hamming weight three, which is derived from a special reduction via Hamming weight three polynomials produced by these two HFE polynomials. This allows us to build new candidates for multiva...

Controlling the peak-to-mean envelope power ratio (PMEPR) of Orthogonal Frequency Division Multiplexed (OFDM) transmissions is a notoriously di cult problem, though one which is of vital importance for the practical application of OFDM in low-cost applications. The utility of Golay complementary sequences in solving this problem has been recognised for some time. In this paper, a powerful theor...

The concepts of Bernoulli numbers B n , Bernoulli polynomials B n (x), and the generalized Bernoulli numbers B n (a, b) are generalized to the one B n (x; a, b, c) which is called the generalized Bernoulli polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between B n , B n (x), B n (a, b), and B n (x; a, b, c) are established.

in this paper we establish several polynomials similar to bernstein's polynomials and several refinements of  hermite-hadamard inequality for convex functions.