Metrics on the Sets of Nonsupersingular Elliptic Curves in Simplified Weierstrass Form over Finite Fields of Characteristic Two

Elliptic curves have a wide variety of applications in computational number theory such as elliptic curve cryptography, pairing based cryptography, primality tests, and integer factorization. Mishra and Gupta (2008) have found an interesting property of the sets of elliptic curves in simplified Weierstrass form (or short Weierstrass form) over prime fields. The property is that one can induce metrics on the sets of elliptic curves in simplified Weierstrass form over prime fields of characteristic greater than three. Later, Vetro (2011) has found some other metrics on the sets of elliptic curves in simplified Weierstrass form over prime fields of characteristic greater than three. However, to our knowledge, no analogous result is known in the characteristic two case. In this paper, we will prove that one can induce metrics on the sets of nonsupersingular elliptic curves in simplifiedWeierstrass form over finite fields of characteristic two.


Introduction
Elliptic curves have been studied by number theorists for a very long time.Nowadays, elliptic curves have been the focus of much attention due to not only the theoretical aspects but also the practical aspects in computational number theory.In particular, elliptic curves have a wide variety of applications in computational number theory such as elliptic curve cryptography [1,2], pairing based cryptography [3,4], primality tests [5,6], and integer factorization [7,8].
Mishra and Gupta in [9] have found an interesting property of the sets of elliptic curves in simplified Weierstrass form (or short Weierstrass form) over prime fields of characteristic greater than three.The property is that one can induce metrics on the sets of elliptic curves in simplified Weierstrass form over prime fields of characteristic greater than three.Later, Vetro in [10] has found some other metrics on the sets of elliptic curves in simplified Weierstrass form over prime fields of characteristic greater than three.They have proposed potential applications of the metrics to the protection of side channel attacks [11].However, to our knowledge, no analogous result is known in the characteristic two case.In this direction, it seems mathematically natural to explore a methodology for constructing metrics on (sub)sets of elliptic curves over finite fields of characteristic two whether there is a cryptographic application or not.
The motivation of this work is to study the characteristic two case.We will prove that one can induce metrics on the sets of nonsupersingular elliptic curves in simplified Weierstrass form over finite fields of characteristic two.
The rest of this paper is organized as follows.In Section 2, we recall some basic facts that will be used throughout the paper.In Section 3, we give metrics on the sets of nonsupersingular elliptic curves in simplified Weierstrass form over finite fields of characteristic two.Section 4 concludes the paper.

Mathematical Preliminaries
In this section we fix our notation and recall some basic facts that will be used throughout the paper.For more details, we refer the reader to [12, Section 3.3], [13, Appendix A].

International Journal of Mathematics and Mathematical Sciences
Let  be a field.For any field , we denote by  = char() the characteristic of the field .We use the symbols Z, R, and F  to represent the integers, real numbers, and a finite field with  elements, where  =   ( ≥ 1),  = char(F  ).For a finite set , we denote the cardinality of  by ♯.
Two elliptic curves  1 and  2 are called isomorphic if there exist morphisms (as algebraic varieties) from  1 to  2 and from  2 to  1 which are inverses of each other.The following theorems (Theorems 2 and 3) tell us when two elliptic curves are isomorphic.
Theorem 4 is the famous Hasse bound for the number of rational points on elliptic curves over finite fields.

Theorem 4 (Hasse). Let ♯𝐸(F
is an elliptic curve with char() = 2 and () ̸ = 0, then the admissible change of variables transforms  to the nonsupersingular elliptic curve An elliptic curve of form ( 5) is called simplified Weierstrass form (or short Weierstrass form).
be nonsupersingular elliptic curves over F 2  in simplified Weierstrass form.If  1 /F 2  ≅  2 /F 2  , then we have  6 =  6 and the isomorphism is given by where  is an element in F 2  and satisfies the equation (see [12,Section 3.3]).

Metrics
In this section we assume that  = F 2  for  ≥ 2. We consider the set of nonsupersingular elliptic curves over F 2  in simplified Weierstrass form; namely, Throughout this section, we assume that   /F 2  ∈ E() are elliptic curves for  = 1, 2, 3.In addition, we denote where 1 ∈ F 2  is the multiplicative identity element.Note that the set B is a nonempty finite set because a polynomial basis belongs to the set B. We choose a basis k = ⟨k 1 , . . ., k  ⟩ ∈ B and fixed it.Then there exists  0 ∈ {1, . . ., } such that k  0 = 1.Let denote the surjective F 2 -linear map, where Note that, for all  ∈ F 2  , we always have wt(  0 ()) ≤  − 1.
We are ready to state and prove the main result of this paper, namely, Theorem 5, which states that the set of nonsupersingular elliptic curves over F 2  in simplified Weierstrass form is a metric space under the metric  ()  k .
Proof.We prove the nonnegativity, the nondegeneracy, the symmetry, and the triangular inequality.
In the definition of  () k , we put  () k ( 1 ,  2 ) =  when  1 /F 2  ≇  2 /F 2  .However, the value  does not have any special meanings, and one can use any other positive integer greater than or equal to  in order to define different metrics on E().
Corollary 7 (other metrics on E()).(1) For any integer  ∈ Z greater than or equal to , define the function  ()  k : E() × E() → R as follows: Then (E(),  () k ) is a metric space.(2) We define the function  (∞)  k : E()×E() → R∪{∞} as follows: Then We choose a subset S ⊆ B. Since B is a finite set, the subset S is also a finite set.For each k ∈ S, we take  k ∈ N  .Then the function is also a metric on E().
Proof.The proofs are very similar to the proof of Theorem 5; thus we omit them.
Remark 8 (topological properties of a metric space E()).
We recall that a metric space gives rise to a topology.Here, we make sure of the properties of the topology on E() induced by a metric.Given a metric space (E(), ), let O  be the topology on E() induced by the metric .Note that the facts shown below do not depend on the metric .A metric space is a Hausdorff space [14, p. 110

Conclusion
In this paper, we have defined some metrics on the sets of nonsupersingular elliptic curves in simplified Weierstrass form over finite fields of characteristic two.In order to derive analogous results for the case of supersingular elliptic curves of characteristic two and for the case of elliptic curves of characteristic three, some deep observation on the properties of elliptic curves over finite fields will be needed.