Elliptic Curve Cryptography Clause Samples

Elliptic Curve Cryptography. Elliptic curve cryptography makes use of elliptic curves in which the variables and coefficients are all restricted to elements of a finite field. Typically, elliptic curves are defined over either the integers modulo a prime number (GF(p)) or over binary polynomials (GF(2m)). An elliptic curve is a cubic equation of the form: y2 +axy + by = x3+ cx2 + dx + e. (1) where a, b, c, d, and e are real numbers. In an elliptic curve cryptosystem (ECC), the elliptic curve equation is defined as the form of Ep(a, b): y2 = x3+ax+b( mod p) (2) over a prime finite field Fp, where a, b ε Fp, p > 3, and 4a3 + 27b2 (mod p) ≠ 0. Generally, the security of ECC relies on the difficulties of the following problems [10]. Definition 1 Given two points P and Q over Ep(a, b), the elliptic curve discrete logarithm problem (ECDLP) is to find an integer s ε Fp* such that Q = s.P. Definition 2 Given three points P, s.P, and t.P over Ep(a, b) for s; t ε Fp*, the computational ▇▇▇▇▇▇-▇▇▇▇▇▇▇ problem (CDHP) is to find the point (s.t).P over Ep(a, b). Definition 3 Given two points P and Q = s.P + t.P over Ep(a, b) or s; t ε Fp*, the elliptic curve factorization problem (ECFP) is to find two points s. P and t.P over Ep(a, b). Up to now, there is no algorithm to be able to solve any of the above problems [10]

Elliptic Curve Cryptography. Elliptic Curve Cryptography (ECC) was discovered in 1985 by ▇▇▇▇▇▇ ▇▇▇▇▇▇(IBM) and ▇▇▇▇ ▇▇▇▇▇▇▇ (University of Washington) as an alternative mechanism for implementing public-key cryptography. The primary benefit promised by ECC is a smaller key size, reducing storage and transmission requirements i.e., an elliptic curve system could provide the same level of security afforded by an RSA-based system with a large modulus and correspondingly larger key e.g., a 256-bit ECC public key should provide comparable security to a 3072-bit RSA public key. In ECC non-singular type of Elliptic curves over the real number are used. The elliptic curve over real numbers takes the general form as: In cryptography, variables and coefficients of elliptic curve equation are restricted to elements in a finite field. Thus for above equation x, y are co-ordinates of GF(p), and are integer modulo , satisfying (for non singular elliptic curve).Where is a modular prime integer which make the EC of finite field. An elliptic curve E over GF(p) consist of points ( ) defined by above two equations, along with an additional point called O (point at infinity or zero point) in EC. The 'O' point plays the role of identity element for EC group. Usually an elliptic curve is defined over two types of finite fields: the prime field containing elements (prime curve) and the characteristic 2 finite field containing 2m elements (binary curve).This paper focuses on the prime finite field as the prime curve are best suit for software applications [22]. Elliptic Curve Arithmetic Cryptographic schemes based on ECC rely on scalar multiplication of elliptic curve points. Given an integer k and a point P ( ), scalar multiplication is the process of adding P to itself k times. The result of this scalar multiplication is denoted k P or kP. Points addition and point doubling form the basis to calculate EC scalar multiplication efficiently using the addition rule together with the double-and-add algorithm or one of its variants. The detail description of ECC (including its point addition rule) can be found in various papers including [12, 20, 26]. The security of ECC based protocols are based on intractability of Elliptic Curve Discrete Logarithm Problem(ECDLP). ECDLP state that: Given P, Q E , find an integer k Z* such that Q = kP. It is relatively easy to calculate Q given k and P, but it is relatively hard to determine k given Q and P.

Elliptic Curve Cryptography. The notion of elliptic curve cryptography was first introduced independently by ▇▇▇▇▇▇ [13] and ▇▇▇▇▇▇▇ [10]. Since then, numerous elliptic curve cryptosystems have been proposed and employed. The main attraction of it is that it allows much smaller parameters (e.g. key size) to be employed in order to achieve an equivalent level of security as compared to the traditional public-key cryp- tosystems such as RSA and DSA. Since an elliptic curve cryptosystem, that is mainly based on the intractability of Elliptic Curve Discrete Logarithm Problem (ECDLP), takes full exponential time, its resistance against the sub-exponential attack offers potential reductions in processing power and memory size which is essential in applications on constrained devices [15]. Let E(Fq) be an elliptic curve of defined over a finite field Fq of characteristic

Elliptic Curve Cryptography. ‌ There are no other techniques to agree on a shared key over a public channel than to use the public-key cryptog- raphy. However, public-key cryptography is costly to be applied in MANETs because of the limited device perfor- ▇▇▇▇▇. Elliptic curve cryptography (ECC) is considered to be most applicable for mobile devices, because of the smaller key sizes ( 160 bits) and more efficient computa- tion compared to other public-key systems. In [10] ECC has been used in the CGKA protocols µSTR and µTGDH in or- der to reduce the communication, computation and mem- ory costs of the original protocols STR and TGDH, respec- tively. Operations of ECC are performed in groups of points of elliptic curves defined over finite fields. In the following we give a brief overview of ECC and elliptic curve ▇▇▇▇▇▇- ▇▇▇▇▇▇▇ key exchange protocol (ECDH). Let E be an elliptic curve over a finite field Fq, such that Fq is either prime (q is a prime numer) or binary (q = 2m, m N) field. E(Fq) denotes a commuta- tive group of points in E. Considering G E(Fq) as a point with high prime order t that devides q 1, there ex- ists a subgroup of points of E(Fq) generated by G, i.e., <G>= O, G, 2G,. . . , (t 1)G , where O is the point of infinity. The operation Q = rP , where r [1, t 1] and P < G > is called a scalar-point multiplication and its result Q is a point in <G>. Note that due to the hard- ness of the Discrete Logarithm problem on elliptic curves (ECDL) it is computationally hard to compute r given P Σm—1 i( ) = 2 ECDH Assume two participants M1 and M2 wish to agree and Q [12]. The protocols µSTR, µTGDH and our frame- work require to map a point Q < G > to an integer in the range [1, q 1]. The most natural way is to map Q to its x-coordinate, denoted (Q)x. The following function map : E(Fq) N ([14]) can be used for this purpose: if q = p and p is prime then map(Q) = (Q)x, else if q = 2m, m ∈ N, and (Q)x = (am—1 . . . a1a0) with ai ∈ {0, 1} then map Q ai. defined h. There are two kinds of nodes: intermediate and leaf nodes. The leaf node is associated with a member (de- vice) and has no children. An intermediate node ⟨l, v⟩ has two children: the left child ⟨l + 1, 2v⟩ and the right child members. Every k(l,v⟩ , 0 ≤ v ≤ 2l − 1, 0 ≤ l ≤ h is com- ⟨l + 1, 2v + 1⟩. In the following we describe the structure of the group key in ECC. Each node ⟨l, v⟩ is associated with a secret key k(l,v⟩ and a public key bk(l,v⟩ = k(l,v⟩G. Secret keys of leaf nodes are chosen and kept secret by associate...

Elliptic Curve Cryptography. Elliptic Curve Cryptography (ECC) The Elliptical Curve Cryptography (ECC) was introduced by ▇▇▇▇▇▇ ▇. ▇▇▇▇▇▇ and ▇▇▇▇ ▇▇▇▇▇▇▇ within the 80s, but it had been only within the late 90's that it began its application. ECC is predicated on the usage of finite field elliptic curves. Moreover, a sum operation is defined 6 over an elliptical curve and a special item O is additionally defined because the identity element. Then, some extent G of the elliptical curve is defined as a base point and an elliptical curve is utilized for the sum operation of a number of elements (points). Elliptical curve cryptosystems are predicated on the intractability of certain mathematical issues, even as the other public key system. Specifically, ECC is predicated on the ECDLP problem that asserts that it's inoperative to compute the discrete logarithm of a random elliptical curve in respect to a base point of an elliptical curve. Elliptic curve cryptography could be a key-based technique for encrypting data. ECC focuses on pairs of public and personal keys for decryption and encryption of web traffic

Elliptic Curve Cryptography. The notion of elliptic curve cryptography was rst introduced independently by ▇▇▇▇▇▇ [13] and ▇▇▇▇▇▇▇ [10]. Since then, numerous elliptic curve cryptosystems have been proposed and employed. The main attraction of it is that it allows much smaller parameters (e.g. key size) to be employed in order to achieve an equivalent level of security as compared to the traditional public-key cryp- tosystems such as RSA and DSA. Since an elliptic curve cryptosystem, that is mainly based on the intractability of Elliptic Curve Discrete Logarithm Problem (ECDLP), takes full exponential time, its resistance against the sub-exponential attack o ers potential reductions in processing power and memory size which is essential in applications on constrained devices [15]. Let E(Fq) be an elliptic curve of de ▇▇▇ over a nite eld Fq of charac- teristic p. The public elliptic curve domain parameters over Fq is de ▇▇▇ as a 8-tuple (q, FR, S, a, b, P, n, h), where q is the prime order of the eld, FR ( eld representation) indicates the representation used for the elements of Fq, S is the random seed for elliptic curve generation, the ▇▇▇ cients a, b R Fq de ne the equation of elliptic curve E over Fq (y2 = x3 + ax + b for p = q > 3, where 4a3 +27b2 = 0), the base point P = (xp, yp) in E(FQ), the prime n is the order of P (n > 2160) and the cofactor h = ]E(FQ)/n, where ]E(FQ) denotes the number of FQ-rational points on E. These parameters should be chosen appropriately to prevent the employment of any e cient algorithm from solving the Discrete Log- arithm Problem (DLP) or the computational Di e-▇▇▇▇▇▇▇ Problem (CDHP) in the cyclic subgroup ⟨P⟩ [16]. Since many cryptographic primitives base their security on the underlying assumptions in which the DLP and CDHP on some cyclic groups are intractable, our proposed key agreement protocol would not be exceptional. Our protocol also rests upon a few related conjectures before its security can be claimed. Now, we de ne several cryptographic problems which we will assume their hardness throughout this paper. Conjecture 1 (ECDLP). Let E(Fq) and P be de ▇▇▇ as above. The Elliptic Curve Discrete Logarithm Problem is said to be intractable if for any proba- blilistic polynomial time Turing Machine A with the knowledge of Y = xP , where Y ∈ ⟨P⟩, the probability of success in computing logP Y = x ∈R P,E(F )q [1, n − 1], denoted as Succecdlp (A) is negligible: Succecdlp P,E(Fq ) (A) = Pr 