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

Abstract

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 simplified Weierstrass form over finite fields of characteristic two.

1. 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.

2. 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].

Let be a field. For any field , we denote by the characteristic of the field . We use the symbols , , and to represent the integers, real numbers, and a finite field with elements, where (), . For a finite set , we denote the cardinality of by .

Let be an algebraic curve over . We setThe next theorem provides a necessary and sufficient condition that is an elliptic curve.

Theorem 1 (see [12, Theorem 2.4]). is an elliptic curve; that is, the Weierstrass equation is nonsingular, if and only if .

Two elliptic curves and are called isomorphic if there exist morphisms (as algebraic varieties) from to and from to which are inverses of each other. The following theorems (Theorems 2 and 3) tell us when two elliptic curves are isomorphic.

Theorem 2 (see [12, Theorem 2.5]). If two elliptic curves and are isomorphic over , then . The converse is also true if is an algebraically closed field.

Theorem 3 (see [12, Theorem 2.1]). Two elliptic curves and given by the equationsare isomorphic over (or -isomorphic), denoted by , if and only if there exists , , such that the change of variablestransforms equation to equation . The relationship of isomorphism is an equivalence relation.

Theorem 4 is the famous Hasse bound for the number of rational points on elliptic curves over finite fields.

Theorem 4 (Hasse). Let . Then .

The elliptic curve is called supersingular if divides . Otherwise, the curve is called nonsupersingular (or ordinary). It is well-known that if or , then is supersingular if and only if . In other words, if or , is nonsupersingular if and only if . Remark that if is an elliptic curve with and , then the admissible change of variablestransforms to the nonsupersingular elliptic curveAn elliptic curve of form (5) is called simplified Weierstrass form (or short Weierstrass form). Let be nonsupersingular elliptic curves over in simplified Weierstrass form. If , then we have and the isomorphism is given bywhere is an element in and satisfies the equation(see [12, Section 3.3]).

Remark that if is a solution of (8) then is the other solution. Furthermore, is an automorphism if and only if ([13, Appendix A, Proposition 1.2]).

3. Metrics

In this section we assume that for . We consider the set of nonsupersingular elliptic curves over in simplified Weierstrass form; namely,Throughout this section, we assume that are elliptic curves for . In addition, we denoteif , , and , respectively. Let be the set of all basis of a linear space over . We putwhere is the multiplicative identity element. Note that the set is a nonempty finite set because a polynomial basis belongs to the set . We choose a basis and fixed it. Then there exists such that . Let denote the surjective -linear map, where (). For any (), we write Note that, for all , we always have . We put (), (), and () for the isomorphisms of form (10).

Now we define the function as follows: Remark that the function is well-defined because, for two solutions , () of (8), we have . Namely, the function does not depend on the choice of isomorphisms.

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 in simplified Weierstrass form is a metric space under the metric .

Theorem 5 (metric on ). is a metric space.

Proof. We prove the nonnegativity, the nondegeneracy, the symmetry, and the triangular inequality.

(1) Nonnegativity. By the definition of the function , we have for all .

(2) Nondegeneracy. Suppose that . Since , we must have . This implies that for all . Remember that . Setting (resp., ) yields (resp., ). When or , the isomorphism of form (10) is an automorphism. Thus, . Conversely, if , then the isomorphism of form (10) is an automorphism. It then follows that . Hence for all and therefore .

(3) Symmetry. If , then . Otherwise, we defineOne can easily check that is an isomorphism. Thus .

(4) Triangular Inequality. Let . We claim thatThere are two cases to consider. Namely, and .

Case 1 (). In this case, we have . If follows immediately from that or . Then or . This shows that as claimed.

Case 2 (). There are two possibilities: and . The former case gives , which shows that (or, more precisely, since ). In the latter case, we have . This yields that , , and , respectively. Since for all , , we obtainwhere (resp., ) is equal to when (resp., ) and otherwise (resp., ) is equal to . Hence we have . This completes the proof.

Remark 6. The main observation of Theorem 5 is that the isomorphism of form (10) is an automorphism if and only if . By omitting the th entry of and by summing up the number of nonzero with and , one can construct a metric on . In order to omit the th entry, we use a basis which is belonging to the set . The nonnegativity and the symmetry for are obvious. The nondegeneracy for is followed by the omission of th entry. The triangular inequality for the Hamming distance implies the triangular inequality for .

In the definition of , we put when . 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 .

Corollary 7 (other metrics on ). (1) For any integer greater than or equal to , define the function as follows: Then is a metric space.
(2) We define the function as follows: Then is also a metric space.
(3) SetWe choose a subset . Since is a finite set, the subset is also a finite set. For each , we take . Then the function is also a metric on .
(4) We put . For each , we take Then as in (3), we can define the metric on .

Proof. The proofs are very similar to the proof of Theorem 5; thus we omit them.

Remark 8 (topological properties of a metric space ). We recall that a metric space gives rise to a topology. Here, we make sure of the properties of the topology on induced by a metric. Given a metric space , let be the topology on 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, Proposition 11.5]. A finite subset of a topological space is compact [14, p. 128, Example 13.8(a)]. These two facts indicate that is a compact Hausdorff space. If a finite space is Hausdorff then its topology is discrete [14, p. 111, Exercise 11.2(b)]. Therefore is discrete, and hence is totally disconnected. Any finite metric space is compact, therefore is a complete metric space.

4. 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.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by a grant for young researchers from Shimane University in 2015.