Let f(x) = ax+bx+c ∈ Z[x] be a quadratic polynomial with a ̸≡ 0 mod p. Take z ∈ Fp and let Oz = {fi(z)}i∈Z+ be the orbit of z under f , where fi(z) = f(fi−1(z)) and f0(z) = z. For M < |Oz|, We study the diameter of the partial orbit OM = {z, f(z), f2(z), . . . , fM−1(z)} and prove that there exists c1 > 0 such that diam OM & min { Mp c1 , 1 log p M 4 5p 1 5 ,M 1 13 log logM } . For a complete or...

Optimal Prime Fields (OPFs) are considered to be one of the best choices for lightweight elliptic curve cryptography implementation on resource-constraint embedded processors. In this paper, we revisit efficient implementation of the modular arithmetic over the special prime fields, and present improved implementation of modular multiplication for OPFs, called Optimal Prime Field Coarsely Integ...

For G a finite group and p a prime, a G-p field is a Galois number field K with Gal(K/Q) ∼= G and disc(K) = ±p for some a. We study the existence of G-p fields for fixed G and varying p. For G a finite group and p a prime, we define a G-p field to be a Galois number field K ⊂ C satisfying Gal(K/Q) ∼= G and disc(K) = ±p for some a. Let KG,p denote the finite, and often empty, set of G-p fields. ...