Given a prime number p, we deduce from a formula of Barsky and Benzaghou and from a result of Coulter and Henderson on trinomials over finite fields, a simple necessary and sufficient condition β(n) = kβ(0) in Fpp in order to resolve the congruence B(n) ≡ k (mod p), where B(n) is the n-th Bell number, and k is any fixed integer. Several applications of the formula and of the condition are inclu...

For a prime p and positive integers l < k < h < p with d = (h, k, l, p−1), we show that M , the number of simultaneous solutions x, y, z, w in Zp to xh + yh = zh + wh, xk + yk = zk + wk, xl + yl = zl + wl, satisfies M ≤ 3d(p− 1) + 25hkl(p− 1). When hkl = o(pd2) we obtain a precise asymptotic count on M . This leads to the new twisted exponential sum bound ∣∣∣∣∣ p−1 ∑ x=1 χ(x)e ∣∣∣∣∣ ≤ 3 4 d 1 2...

This work presents novel multipliers for Montgomery multiplication defined on binary fields GF(2). Different to state of the art Montgomery multipliers, this work uses a Linear Feedback Shift Register (LFSR) as the main building block. We studied different architectures for bit-serial and digit-serial Montgomery multipliers using the LFSR and the Montgomery factors x and x. The proposed multipl...

For an infinite family of monogenic trinomials \(P(X)=X^3\pm 3rbX-b\in {\mathbb {Z}}[X]\), arithmetical invariants the cubic number field \(L={\mathbb {Q}}(\theta )\), generated by a zero \(\theta\) \(P(X)\), and its Galois closure \(N=L(\sqrt{d_L})\) are determined. The conductor \(f\) cyclic relative extension \(N/K\), where \(K={\mathbb {Q}}(\sqrt{d_L})\) denotes unique quadratic subfield \(...

The functional decomposition of polynomials has been a topic of great interest and importance in pure and computer algebra and their applications. The structure of compositions of (suitably normalized) polynomials f = g ◦ h in Fq[x] is well understood in many cases, but quite poorly when the degrees of both components are divisible by the characteristic p. This work investigates the decompositi...