On The Elliptic Divisibility Sequences over Finite Fields

In this work we study elliptic divisibility sequences over finite fields. Morgan Ward in [11, 12] gave arithmetic theory of elliptic divisibility sequences. We study elliptic divisibility sequences, equivalence of these sequences and singular elliptic divisibility sequences over finite fields Fp, p > 3 is a prime. Keywords—Elliptic divisibility sequences, equivalent sequences, singular sequences. I. PRELIMINARIES. A divisibility sequence is a sequence (hn) (n ∈ N) of positive integers with the property that hm|hn if m|n. The oldest example of a divisibility sequence is the Fibonacci sequence. There are also divisibility sequences satisfying a nonlinear recurrence relation. These are the elliptic divisibility sequences and this relation comes from the recursion formula for elliptic division polynomials associated to an elliptic curve. An elliptic divisibility sequence (or EDS) is a sequence of integers (hn) satisfying a non-linear recurrence relation hm+nhm−n = hm+1hm−1hn − hn+1hn−1hm (1) and with the divisibility property that hm divides hn whenever m divides n for all m ≥ n ≥ 1. There are some trivial examples such as the sequence of integers Z 0, 1, 2, 3, 4, 5, 6, · · · is an EDS but non-trivial examples abound. The simplest EDS is the sequence 0, 1, 1,−1, 1, 2,−1,−3,−5, 7,−4,−28, 29, 59, 129, −314,−65, 1529,−3689,−8209,−16264, 83331, 113689,−620297, 2382785, 7869898, 7001471, −126742987,−398035821, 168705471, · · · . This is the sequence A006769 in the On-Line Encyclopedia of Integer Sequences maintained by Neil Sloane. EDSs are generalizations of a class of integer divisibility sequences called Lucas sequences, [10]. EDSs were interesting because of being the first non-linear divisibility sequences to be studied. Morgan Ward wrote several papers detailing the arithmetic theory of EDSs [11, 12]. For the arithmetic properties of EDSs, see also [2, 3, 4, 5, 9]. Shipsey and Swart [6, 9] interested in the properties of EDSs reduced modulo primes. The Chudnovsky brothers considered prime values of EDSs in [1]. Rachel Shipsey [5] used EDSs to study Osman Bizim is with the Uludag University, Department of Mathematics, Faculty of Science, Bursa-TURKEY, email: [email protected]. This work was supported by The Scientific and Technological Research Council of Turkey, project no: 107T311. some applications to cryptography and elliptic curve discrete logarithm problem (ECDLP). EDSs are connected to heights of rational points on elliptic curves and the elliptic Lehmer problem. A solution of (1) is proper if h0 = 0, h1 = 1, and h2h3 = 0. Such a proper solution will be an EDS if and only if h2, h3, h4 are integers with h2|h4. An EDS which do not satisfy one (or more) of these conditions is called improper elliptic divisibility sequence. The sequence (hn) with initial values h1 = 1, h2, h3 and h4 is denoted by [1 h2 h3 h4]. An integer m is said to be a divisor of the sequence (hn) if it divides some term with positive suffix. Let m be a divisor of (hn). If ρ is an integer such that m|hρ and there is no integer j such that j is a divisor of ρ with m|hj then ρ is said to be rank of apparition of m in (hn). Elliptic divisibility sequences are a generalization of a class of divisibility sequences studied earlier by Edouard Lucas. In fact many of Ward’s results about EDSs were prompted by similar results discovered by Lucas for his sequences. Let α be a rational number, and let a and b the roots of the polynomial x − αx+ 1. If a = b let (ln) be the sequence ln = a − b a− b for n ∈ Z. If a = b define ln = nan−1. Then (ln) is called a Lucas sequence with parameter α. Ward said that the Lucas sequence (ln) is an EDS if and only if α is an integer. Lucas sequences are special case of a type of EDS called a singular EDS. The following definition will show us that which EDSs are singular. Discriminant of an elliptic divisibility sequence (hn) is defined by the formula Δ (h2, h3, h4) = 1 h2h 3 3 ⎡ ⎣ (h 4 4 + 3h 5 2h 3 4 + (3h 8 2 + 8h 3 3)h 2 4 +h2(h 8 2 − 20h3)h4 +h2h 3 3(16h 3 3 − h2) ⎤ ⎦ . An elliptic divisibility sequence (hn) is said to be singular if and only if its discriminant Δ(h2, h3, h4) vanishes. Now we see that when two EDSs are equivalent so we need to know following definition: Definition 1.1: Two elliptic divisibility sequences (hn) and (hn) are said to be equivalent if there exists a constant θ such that hn = θ n2−1hn for all n ∈ Z. World Academy of Science, Engineering and Technology 35 2009