Elliptic Curve Cryptography (ECC) is a standout amongst the most interested exploration themes in VLSI. System security is turning out to be increasingly significant as the volume of information being traded on the Internet increments. Point addition and doubling are key operations which choose the Performance of ECC. Here the design with the information way which can perform either prime field...

Let Fp be the field of prime order p. It is known that for any integer N ∈ [1, p] one can construct a subset A ⊂ Fp with |A| = N such that max{|A+ A|, |AA|} p|A|. One of the results of the present paper implies that if A ⊂ Fp with |A| > p2/3, then max{|A+ A|, |AA|} p|A|.

For each finite field F of prime order there is a constant c such that every 4-connected matroid has at most c inequivalent representations over F. We had hoped that this would extend to all finite fields, however, it was not to be. The (m,n)-mace is the matroid obtained by adding a point freely to M(Km,n). For all n ≥ 3, the (3, n)-mace is 4-connected and has at least 2 representations over an...

Let Fp be the field of residue classes modulo a prime number p and let A be a nonempty subset of Fp. In this paper we show that if |A| p , then max{|A ± A|, |AA|} |A|; if |A| p, then max{|A ± A|, |AA|} v min{|A|( |A| p0.5 ), |A|( p |A| )}. These results slightly improve the estimates of Bourgain-Garaev and Shen. Sum-product estimates on different sets are also considered.

The celebrated Weil bound for character sums says that for any low-degree polynomial P and any additive character χ, either χ(P ) is a constant function or it is distributed close to uniform. The goal of higher-order Fourier analysis is to understand the connection between the algebraic and analytic properties of polynomials (and functions, generally) at a more detailed level. For instance, wha...