Elliptic Curve over a Local Finite Ring Rn

The goal of this chapter is to study some arithmetic proprieties of an elliptic curve defined by a Weierstrass equation on the local ring Rn 1⁄4 q X 1⁄2 = X ð Þ, where n≥ 1 is an integer. It consists of, an introduction, four sections, and a conclusion. In the first section, we review some fundamental arithmetic proprieties of finite local rings Rn, which will be used in the remainder of the chapter. The second section is devoted to a study the above mentioned elliptic curve on these finite local rings for arbitrary characteristics. A restriction to some specific characteristic cases will then be considered in the third section. Using these studies, we give in the fourth section some cryptography applications, and we give in the conclusion some current research perspectives concerning the use of this kind of curves in cryptography. We can see in the conclusion of research in perspectives on these types of curves.


Introduction
Elliptic curves are especially important in number theory and constitute a major area of current research; for example, they were used in Andrew Wiles's proof of Fermat's Last Theorem.They also find applications in elliptic curve cryptography (ECC), integer factorization, classical mechanics in the description of the movement of spinning tops, to produce efficient codes … For these reasons, the subject is well known, presented, and worth exploring.
The purpose of cryptography is to ensure the security of communications and data stored in the presence of adversaries [1][2][3].It offers a set of techniques for providing confidentiality, authenticity, and integrity services.Cryptology, also known as the science of secrecy, combines cryptography and cryptanalysis.While the role of cryptographers is to design, build, and prove cryptosystems, among other things, the goal of cryptanalysis is to "break" these systems.The history of cryptography has long been the history of secret codes and along all previous times, this has affected the fate of men and nations [4].In fact, until 1970, the main goal of cryptography was to build a signature encryption systems [5,6], but thanks to cryptanalysis, the army and the black cabinets of diplomats were able to wage their wars in the shadows controlling the communication networks, especially of their enemies [7,8].The internet revolution and the increasingly massive use of information in digital form facilitated communications but in counterparty it weakened the security level of information.Indeed, "open" networks create security holes, which allow access to the information.Cryptography, or the art of encrypting messages, a science that sites today in the crossroads of mathematics, computer sciences, and some applied physics, has then become a necessity for today's civilization to keep its secrets from adversaries.Confusion is often made between cryptography and cryptology, but the difference exists.Cryptology is the "science of secrecy," and combines two branches on the one hand, cryptography, which makes it possible to encrypt messages, and on the other hand, cryptanalysis, which serves to decrypt them.Our focus in this chapter is to show how some elliptic curves, mathematical objects studied particularly in algebraic geometry [9][10][11][12].You can give several definitions depending on the person you are talking to.Cryptography indeed used elliptic curves for more than 40 years the appearance of the Diffie-Hellman key exchange protocol and the ElGamal cryptogram [13][14][15].These cryptographic protocols use in particular group structures, for by applying these methods to groups defined by elliptic curves, a new speciality was born at the end of the 1980: ECC, Elliptic Curve Cryptography.Recall that Diffie-Hellman key exchange which is based on the difficulty of the discrete logarithm problem (DLP) [16][17][18].The success of elliptic curves in public key cryptographic systems has then created a new interest in the study of the arithmetic of these geometric objects.The group of points on an elliptical curve is an interesting group in cryptography because there is no known sub-exponential algorithm for sound (DLP) [19][20][21].In general, the DLP is difficult to be solved, but not as much as in a generic group as in the case of finite field.We know sub-exponential algorithms to solve it depending on the size of the group to use, which impose criteria for the PLD to be infeasible.The prime number p which is the characteristic of our base ring must then have at least 1024 bits, which offers a security level similar to the one given by a generic order group of 160 bits.Recall that a generic group for the DLP is a group for which there is no a specific algorithm to solve the DLP [22], so that the only available algorithms are those for all groups.
In [23], Elhassani et al. have built an encryption method based on DLP and Lattice.Boulbot et al. in [24] have studied elliptic curves on a non-local ring to compare these curves on local and non-local rings, while in [25], Sahmoudi et al. have studied these types of curves on a family of finite rings in the authors have introduced a cryptosystem on these types of curves, see [26].
In this chapter, d and n are a positive integers and q ¼ p d is a power of a prime natural number p.

Internal laws in R n
Recall that the two laws "+" and "." are naturally defined on R n [30,31]: for every two elements X ¼ P nÀ1 i¼0 x i ϵ i and Y ¼ 2

Proof.
By formula (2), we have then, x 0 2kþ1 ¼ 2 Under the same hypotheses of the corollary (2.1) and by an analogous proof, we have the following corollary: Corollary 2.2 X 3 ¼ P nÀ1 i¼0 x 00 i ϵ i , where

Proof.
Let Y ¼ P nÀ1 i¼0 y i ϵ i be the inverse of X ¼ which means that, Lemma 2. 4 The non inverse elements in R n are the elements of the form 3), X is invertible in R n if and only if x 0 is invertible in  q : As  q is a field, this means x 0 6 ¼ 0.
Corollary 2. 5 The ring R n is local, with maximal ideal I n ¼ ϵR n .Notation.
Corollary 2.6 π k et k π are two ring homomorphisms.

Proof.
We have, and 4

Number Theory and Its Applications
Note that in addition, for every k ≥ 1, So, π k and k π are tow rings morphisms.Theorem 2.7 Let n ≥ 2 be an integer, where, and Proof.

Primitive triples
Definition 2.8 Let R be a ring.We say that an element x, y, z The set of these primitive triplets will be denoted P R ðÞ .Remark 2.9 The equality xR þ yR þ zR ¼ R means that there exists α, β, λ Proposition 2.10 Let R be a local ring, then x, y, z ðÞ ∈ R 3 is a primitive triple if and only if at least one of the elements x, y, and z is invertible in R.
Proof.Suppose that x, y and z are not invertible in R, then: x, y, z ðÞ ∈ M 3 where M is the unique maximal ideal of R, hence which contradicts that x, y, z ðÞ is a primitive triple.Conversely, suppose, for example, that x is invertible in R, Remark 2.11 If R is a field, then an element x, y, z ðÞ ∈ R 3 is primitive if and only if x, y, z ðÞ 6 ¼ 0, 0, 0 ðÞ .

The projective plane on a finite ring
Let R is a ring.The projective plane on R is the set of equivalence classes of P R ðÞ modulo; the equivalence relation $ R defined by: We denote the projective plane on R by  2 R ðÞ , it is the quotient set P R ðÞ $R , and we write x : y : z ½ for the equivalence class of x, y, z ðÞ ∈ P R ðÞ .Thus, we have: Example 2.12 We Consider the finite ring , where e is an indeterminate satisfying e 2 ¼ 0. The group of units for this ring is

Elliptic curve over R n
In this section, we study the elliptic curves defined on finite local rings R n of characteristic a prime number p; 1.A projective Weierstrass equation on R n is an equation of the form: 2. A affine Weierstrass equation on R n is an equation of the form: where a 1 , a 2 , a 3 , a 4 , a 6 ðÞ ∈ R n 5 :

Elliptic curve form
To an affine (or projective) Weierstrass Eqs. ( 3) and ( 4), we associate the following quantities: Elliptic Curve over a Local Finite Ring R n DOI: http://dx.doi.org/10.5772/intechopen.93476 Δ is called the discriminant of E and j its j -invariant.Remark 3.1 On the field R 1 ¼  q ,wedenotethediscriminantbyΔ 0 and the j-invariant by j 0 , while on the ring R n ,n> 1 we denote the discriminant by Δ ε,n and the j-invariant by j ε,n .
Definition 3.2 Let R be a finite ring and let a ¼ a 1 , a 2 , a 3 , a 4 , a 6 ðÞ ∈ R 5 : An elliptic curve on R corresponding to a, which we write E a R ðÞ , is the set of zeros in the projective plane  2 R ðÞ of the Weierstrass Eq. ( 3), for which the discriminant Δ is invertible in R.

Remark 3.3 According to the characteristic of the ring R; chra R
ðÞ we have the following cases: ðÞ has one of the following forms: for a ¼ a 1 , a 3 , a 4 , a 6 ðÞ ∈ R 4 , with a 1 non invertible and ðÞ has one of the following forms: Remark 3.4 A projective elliptic curve on a field K has one of the following normal forms (Table 1):

Projective coordinates and group law
In this subsection, we give in projective coordinates the formulas for adding the points on an elliptic curve defined by Eq. ( 3) on the ring R n , according to the normal form.
Proof.π k is well defined because π k is a morphism of rings.According to theorem (2.7), we have and The coefficients A, B and ÀC are the partial derivatives of the function calculated starting from x 0 , y 0 , z 0 ÀÁ , which are not all equal to zero and deducing the existence of x kÀ1 : y kÀ1 : z kÀ1 ÂÃ .Hence, π k est surjectif.Using corollary (2.6), we deduce that π k is a group homomorphism.Lemma 4.2 For all k ≥ 2, Proof.
We have: so, z kÀ1 ¼ 0: is an injective group homomorphism.

Main applications
In this subsection, we consider a prime p which does not divide N, where Proof.

Proof.
Let P ∈ k G , we have pP ¼ 0 : Hence, there is a unique homomorphism which makes the diagram(d) commutative.Theorem 4.7 If p do not divide N, then there exists a unique homomorphism According to Lemma (4.6), there is a unique homomorphism which makes the following diagram commutative; (named Diagram(d')): The isomorphism admits an inverse application ðÞ , NN 0 P ðÞ Indeed, Likewise, 14

Proof.
We have According to Haas' theorem, we have: (97)

Applications
In this section, we are interested in ECC using elliptic curves over the ring R n .

The discrete logarithm on E n a,b
The discrete logarithm problem that we denote DLP, (Discrete logarithm problem), is a generally difficult problem which depends on the considered group G.In many situations, due to the asymmetry existing between problems concerning the calculation of logarithms and calculation of powers which is more easier and so of great interest in cryptograph, the above mentioned makes Diffie and Hellman were the first to build a cryptosystem from this situation [37,38].
Definition 5.1 Let G be a finite cyclic group of order ρ and s, r two elements of G.We call discrete logarithm of base s of r, the only element m in 0, ρ À 1 ½ ½ such that s m ¼ r: The discrete logarithm for elliptic curves is defined in an analogous way to be, the only element m in 0, ρ À 1 ½ ½ such that mP ¼ Q, where P, and Q are two points of an additive subgroup G of E n a,b : By using the isomorphism proved given in theorem (4.8), we get the results gathered in the next theorem: Theorem 5.2 If p does not divide N, then.
• #E n a,b ¼ p dn À1 ðÞ Â N: • The problem of the discrete logarithm on the elliptic curve E n a,b is equivalent to that of E 1 a 0 ,b 0 : • If the problem of the discrete logarithm on E n a,b is trivial, then it is also trivial on the elliptic curve E 1 a 0 ,b 0 :

Cryptography based on elliptic curves E n a,b
Elliptic curve cryptography (ECC) is public key cryptography, which relies on the use of curves over finite fields.Essentially, there are two families of these curves which are used in cryptography.The first uses elliptic curves on a finite field  p d , where p is a large prime number.This family is the best choice for a high software level when implementing ECC.The second family uses elliptic curves on a binary field  2 d where d is a large positive integer, this family is more appropriate at the material level point of view when implementing ECC.Another family which is also interesting in ECC implementations is the family of elliptic curves on the previously seen rings R n .The most important advantage presented by the use of elliptic curves in cryptography (ECC) consists in the high security they provide for wireless applications compared to other asymmetric key cryptosystems, also their small key size.Indeed, a 160-bit key for (ECC) can replace a 1024-bit key for (RSA).Given d; a large integer, where This is in fact a difficult problem, whose resolution is exponential.

Elliptical Diffie-Hellman cryptosystem
Recall that Alice and Bob can publicly agree on a common secret (that we describe below).2. As soon as we have a group E n a,b , and an element P ∈ E n a,b of finite order we can consider a Diffie-Hellman system on G¼ < P > which is cyclic.For this construction to have a cryptographic interest, log P tP ðÞ ¼ t must be not easy to calculate.

E n
a,b , is not always cyclical.
4. If, Oscar (program) is giving d, E n a,b , tP and sP, then it is able to solve the discrete elliptical logarithm problem and find tors:

Elliptical ElGamal cryptosystem
Let P m ∈ E n a,b be the point representing the message m, to encrypt P m : 1. Key generation algorithm • Bob chooses the private key t ∈  known only to him.
• d ∈ ℕ, P ∈ E n a,b and R ¼ t ½ P are public.

Encryption algorithm
• Alice Randomly chooses k ∈ ; 3. To decrypt received message c 1 , c 2 ðÞ , Bob calculates: Now, Oscar encounters the discrete elliptic logarithm problem, because to decipher the message P m he must know t (i.e.; calculate t such that R ¼ t ½ P).

Coding example
Let d be a positive integer, we consider the quotient ring , where  2 d is the finite field of order 2 d .Then the ring R 2 is identified with the ring  2 d ε ½ , where ε 2 ¼ 0, i.e., We consider the elliptic curve on the ring R 2 given by the equation: Let E 2 a,b be the elliptic curve over R 2 and consider the irreducible polynomial ½ TX ðÞ ðÞ , then 1, α, α 2 ðÞ is a vector space base of  8 over  2 : ⋆ Put: We have: a,b of order l, and consider the subgroup G¼ < P > , generated by P,t o encrypt and decrypt our messages.

Coding of elements of G
We will give a code to each element Q ¼ mP, where m ∈ 1, 2, ::, l fg , defined as follows: , where x i , y i ∈  8 for i ¼ 0, 1, and Z ¼ 0 or 1, then we set: (106) where α is the primitive root of the irreducible polynomial TX ðÞ So, we code Q as follows: The elliptic curve E 2 a,b contains 112 elements to know: 19 Elliptic Curve over a Local Finite Ring R n DOI: http://dx.doi.org/10.5772/intechopen.93476 We consider: , then G¼ < P > is of order 28.We attach to each point Q ∈ G a letter of the alphabet and a code.We collect the results in the following Table 2: mP Codefor mP Symbol  is: hello to abdelhakim.

Conclusion
The results obtained are very important from theoretical points of view because to study an elliptic curve on a finite local ring it suffices to study these curves on finite fields, for the applications of these curves they can be applied in cryptography to reinforce security and we can use them in cryptanalysis to solve the PDL on special curves.This results are very imploring and give applications in different fields such as classical mechanics, number theory, cryptology, information theory … and we can quote here: 1.The generalization of Hass's theorem, corollary 4.9.
2. The result of the corollary 4.11, then in [24], we have the result of the Proposition 3.12.

1 . 16 Number Theory and Its Applications 3 . 4 . 5 .Remark 5 . 3 1.
They choose on a large integer d, E n a,b and P ∈ E n a,b : 2. Alice chooses t ∈  and calculates tP: Bob chooses s ∈  and calculates sP: Alice let public tP and keep private t: Bob let public sP and keep private s: 6.Then, Alice and Bob build their common secret key K ¼ tsP ¼ stP: Unlike the classic Diffie-Hellman algorithm, we do not ask that P be a generator of E n a,b : The analogue of the subgroup  * p of order p À 1, is the cyclic subgroup of E n a,b , generated by the point P: c 20 c 01 c 11 c 21 d 00 d 10 d 20 d 01 d 11 d 21 1: If Z ¼ 0: Q ¼ 00c 01 c 11 c 21 d 01 d 11 d 21 10000: Number Theory and Its Applications 2. Example: with the same a and b;

5. 6
Encryption and decryption procedures• The encryption of our message "for the elliptical curve", is;

a6 Table 1 .
Elliptic curve form on a field.