SEMINAR SECURITY - REPORT ELLIPTIC CURVE CRYPTOGRAPHY

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1 SEMINAR SECURITY - REPORT ELLIPTIC CURVE CRYPTOGRAPHY OFER M. SHIR, THE HEBREW UNIVERSITY OF JERUSALEM, ISRAEL FLORIAN HÖNIG, JOHANNES KEPLER UNIVERSITY LINZ, AUSTRIA ABSTRACT. The area of elliptic curves has been around for quite a while. The interest in its application for cryptography has been raised only a short time ago and has not become as popular as some of todays well known cryptosystems. This paper should give a brief introduction to that field and its applications. 1. INTRODUCTION 1.1. History. Cryptography, the wide field of security issues in message transmission and storage, reaches back to at least the era of the Roman empire and was of general interest ever since. Until the late seventies, those systems could be classified as what we call private key systems, which have the most significant feature of the shared secret. Since 1976 a lot of number-theoretical and algebraic studies in public key cryptography have arose, since this new class became of strong interest, not requiring a shared secret and therefore can be used in a much more flexible and instant manner. Detailed description of this distinction can be found in a very wide choice of literature. Further on, around 1985, the elliptic curve theory as well as the more general hyperelliptic curve theory have been inspected in the context of various different problems from the field of cryptography, such as: Factorization of integers, primality testing or cryptosystems themselves. The nature of those systems can be seen in an analogous way to the classical number-theoretical problems in cryptography, as will be shown in this section later Brief Algebraic Background. In order to being able to describe and understand the basics of Elliptic Curve Cryptography, we would like to review some mathematics. In the framework of this paper, we assume basic understanding of general algebra, algebraic structures and elementary number-theoretical knowledge. Further we will not give any proofs on the given theorems since those can be found in the literature, and are not a goal of this report. Fields, Generators and Characteristics. Given a field F, with addition and multiplication operators under all the familiar rules, the characteristics char(f) of that field is defined 0 if adding the multiplicative identity I repetitively to itself never yields 0. Otherwise, there exists a prime number p such that I + I I (p times) yields 0, and then p is called the characteristic of the field F. Example. Given the field of rational numbers Q, it is trivial to show that char(q) = 0. Let us introduce a finite field Z/pZ (also denoted as either GF(p) or F p ) as the field of integers modulo a prime number p. It is guaranteed that for every prime power q = p r, there is a unique field of q elements (uniqueness up to an isomorphism). Example. F 5 =< {0,1,2,3,4},+, > Date: April

2 SEMINAR SECURITY - REPORT ELLIPTIC CURVE CRYPTOGRAPHY 2 Further more, let us denote the set of nonzero elements of our finite field as F q. Those q 1 elements form an abelian group, with respect to multiplication. Example. F 5 =< {1,2,3,4}, > Define the order of a nonzero element as the least positive power which yields the identity I. A generator g of a finite field is an element of order q 1. Taking all the powers of g, they run through all the non-zero elements of F q. Every finite field has a generator. Given g, a generator of F q, then g j is also a generator : gcd( j,q 1) = 1. In particular, there are a total of ϕ(q 1) different generators for F q, where ϕ is the Euler phi-function: ϕ(n) = {0 b < n gcd(b,n) = 1} Example. The generator of F 5 is 2. One can easily verify that The Discrete Logarithm Problem. The difficulty of computing logarithms makes exponentiation in a finite field a one-way function, and therefore applicable for cryptography. There are numerous cryptosystems whose security depends on this difficulty of solving the discrete logarithm problem. Let us state now the following problem, based on the last sections definitions: Definition. Let G be a multiplicative abelian group, and for g G, let F q be the cyclic subgroup generated by g, then given g and a F q, find and integer x such that g x = a. Example. In order to gain some intuition, let us look at the simple group F 5 from above where g = 2. If a is for example 3, find x such that 2 x = 3. An integer x which satisfies that would be 3, since 2 3 (mod 5) = 3. However, since we work in a cyclic group, any integer which satisfies that with a remainder of 3 when divided by 5 could be the solution. One can easily see that there is no trivial solution to this problem. The following cryptosystems take advantage of this one-way property. The Diffie-Hellman Key Exchange. To avoid initial confusion of the Diffie-Hellman key exchange algorithm with a public key cryptosystem, we would like to emphasize at this point, that it is merely a secure way to agree on a common shared secret key for a private key communication encryption manner. It works as follows: Suppose that Andrea and Bernhard want to agree on a shared secret key (large integer), in order to communicate with their preferred private key cryptosystem. The agreement is done in open communication channels, such that everybody can read everything that they send and receive. First, they both agree on a large prime p and a generator g of the abelian group F p. Both can be of public knowledge. Second, Andrea secretly chooses a positive random integer, K A < p, in about the same magnitude as p, and sends g K A(mod p) to Bernhard. The latter does the equivalent by sending his g K B(mod p) to Andrea, while keeping his K B secret as well. The agreed upon key will then be: g K AK B. Note that only them can produce this number, since g K AK B = (g K A) gkb = (g K B) gka. The security of this system is broken if the following problem is solved: Given g, g K A, g K B F p, find g K AK B. One can immediately see that if the DLP is solved, this system is broken. The Massey-Omura cryptosystem. The following cryptosystem facilitates secure message transmission between two parties and works as follows: All the users in the system agree on a publicly known finite field F q which is fixed. Each user chooses a secret random integer e, 0 < e < q 1, such that gcd(e, q 1) = 1, and then computes its inverse d = e 1 (mod q 1) using the Euclidean Algorithm. Assume

3 SEMINAR SECURITY - REPORT ELLIPTIC CURVE CRYPTOGRAPHY 3 the user Adeline wants to send a message M to Bert, she sends him the element M e A. Without knowing the inverse d A, it is useless to everybody. Further more, Bert takes it to the power of his d B and sends back M e Ae B to Adeline. She will then raise it to the power of d A, meaning that she reveals her part of the message encryption, leaving M e B - which only Bert has the inverse for - and send it to him. He will take it to the power of d B and retrieve M. The ElGamal cryptosystem. This next cryptosystem described here also provides a secure message transmission between two parties. Let s start with a rather large finite field F q and an element g F q which is preferably, but not necessarily, a generator of that group. In order to transmit a plain text message M, which is mapped to elements of F q, the user Albert chooses a random secret integer a A, 0 < a A < q 1, and publishes g a A. To be able to send a message M to Albert, Benedetta has to choose a secret random integer and send Albert the following pair: (g k, Mg a Ak ). Since Albert knows a A, he can compute g a Ak and divide the second pair-element in order to retrieve M. Solving the DLP can break this system. 2. ELLIPTIC CURVES 2.1. Definitions. Given a field K of characteristic 2,3, and given a cubic polynomial with no multiple roots, x 3 + ax + b (s.t. a,b K), we define an elliptic curve over K as the set of points (x,y) with x,y K which satisfy the following equation: y 2 = x 3 + ax + b together with a single element called point at infinity, denoted O. If K is a field of characteristic 2, then an elliptic curve over K is the set of points satisfying the equation: ( ) y 2 + y = x 3 + ax + b together with a point at infinity O. In this case we don t mind if the polynomial has multiple roots. If K is a field of characteristic 3, then an elliptic curve over K is the set of points satisfying the equation: ( ) y 2 = x 3 + ax 2 + bx + c ( ) together with a point at infinity O, whereas the cubic polynomial has no multiple roots. Let us introduce the fundamental fact on which the entire Elliptic Curve Cryptography field relies on: Theorem. these formulas give an abelian group law on an elliptic curve over any field. Equivalently, the set of points on an elliptic curve over any field form an abelian group. At this point we would like to visualize the group law for the case K = R, where the elliptic curve is an ordinary curve in the plane - along with the point at infinity. Definition. Let E be an elliptic curve over the Real numbers. Given two points on E, denoted P and Q, we define the negative of P and the sum P+Q according to the following rules:

4 SEMINAR SECURITY - REPORT ELLIPTIC CURVE CRYPTOGRAPHY 4 (1) O is the additive identity of the group, or equivalently the zero element. Let P = O, then we define P to be O, and P + Q to be Q. In what follows assume neither P nor Q is the point at infinity. (2) The negative P is the point with the same x-coordinate but with a negative y- coordinate of P: (x,y) (x, y) It follows from the characteristic equation of the curve ( ) (due to symmetry considerations) that (x, y) is on the curve whenever (x,y) is. (3) Given P,Q with different x-coordinates, it can be shown that the line L PQ intersects the curve in exactly one more point R. Then we define P + Q to be R. (4) If Q = P then we define P + Q = O. It is of course forced by (1). (5) If P = Q, then given L, the tangent line to the curve at P, let R denote the only point of intersection of L with the curve - so then we define P + Q 2P = R. Remark. The fact that there exists a unique third point of intersection with the curve follows from trivial analytical geometry. Moreover, the explicit result for the case (3): ( ) y2 y 2 1 x 3 = x 1 x 2 x 2 x 1 ( ) y2 y 1 y 3 = y 1 + (x 1 x 3 ) x 2 x 1 whereas for the case (5) the result is given by: ( 3x 2 ) 2 x 3 = 1 + a 2x 1 2y 1 ( 3x 2 ) y 3 = y a (x 1 x 3 ) 2y 1 A visualization of the addition law in R is given in Figure 2.1. There are few ways to prove that the given definition of P + Q makes the points on an elliptic curve into an abelian group: (1) Projective Geometry [3]. (2) Complex Analysis [4]. (3) Algebra (divisors on curves) [5]. From this point let us discuss only Elliptic Curves over finite fields. A way to construct a curve and a point over it is described in algorithm 1: Algorithm 1 Constructing an Elliptic Curve and a point over it Choose a large finite field F q (assume char(f q ) > 3). Let a,x,y be three random elements of the field. Set b = y 2 ( x 3 + ax ). While (the cubic polynomial x 3 + ax + b has multiple roots) generate new a,x,y. Set B = (x,y), and thus B is a point of E, since y 2 = x 3 + ax + b. Given a finite field F q of q = p r elements, let E be an elliptic curve over F q. One should expect at most N = 2q + 1 points on the elliptic curve - 2q pairs (x,y) (where x,y F q )

5 SEMINAR SECURITY - REPORT ELLIPTIC CURVE CRYPTOGRAPHY 5 FIGURE 2.1. The P+Q Visualization for an EC over the Reals which satisfy the characteristic equation ( ), along with the point at infinity. Let us introduce Hasse s Theorem, which provides an upper bound on the number N of points on the elliptic curve: Theorem. N (q + 1) 2 q There exists a deterministic polynomial-time algorithm for finding the number N of points over an elliptic curve - Schoof s Algorithm. Remark. Although knowing N is not necessary for the implementation of all the cryptosystems, it is important to know it in order to boost the confidence in some cases (regarding the Silver-Pohlig-Hellman attack, on which will be written later on) The Analog Discrete Logarithm Problem. Given an elliptic curve E over F q, the Elliptic Curve analogy of multiplying two elements in F q is adding two points on E. Hence, the analog of raising to the k th power in F q is multiplication of a point P on E by an integer k. Definition. Given E, an elliptic curve over F q, and a point B E - then the discrete log problem on E to the base B is the following problem (the ECDLP): Given a point P E, find an integer x Z such that x B = P, if such an integer exists. We shall describe briefly the three cryptosystems which are analogous to the above cryptosystems given in section 1.3. Analog of the Diffie-Hellman Key Exchange. Suppose E being an elliptic curve over F q, and P an agreed upon point on E which is publicly known. In order to share a secret common key, both Avital and Boaz secretly choose integers K A,K B respectively, and exchange the products K A P, K B P. The common key will be then K A K B P. Without solving the ECDLP, an eavesdropper cannot retrieve this common key. Analog of Massey-Omura. Let P m be an embedded message m as a point on some fixed and publicly known elliptic curve E over F q. Assume that the number N of points on E has been computed, and is also publicly known. Each user in our system selects a random integer e,

6 SEMINAR SECURITY - REPORT ELLIPTIC CURVE CRYPTOGRAPHY 6 such that 1 e N gcd(e,n) = 1 - and computes its inverse d (i.e. d e 1mod N). The transmission protocol works as follows: Algorithm 2 EC Massey-Omura A: compute e A P m and send to B. B: compute e B e A P m and send to A. A: compute d A e B e A P m e B P m and send back to B. B: compute d B e B P m P m, with Pm 1 being the original plain text m. Without solving the ECDLP, one cannot retrieve the original plain text m. Analog of ElGamal. Choose a large finite field F q and an elliptic curve E over it. Let the users of the system choose a point B E. Let Aviezer choose secretly a random integer k a, and publish k a B E. We assume that the message P is again embedded over E. In order to send P to Aviezer, Bracha chooses randomly an integer c, and transmit the pair (c B,P + c (k a B)). At this stage Aviezer will multiply the first element by his key, subtract it from the second element - and get P. Without solving the ECDLP, it is impossible to get P. 3. DISCUSSION 3.1. Comparison. Elliptic Curves were originally introduced with keeping in mind the analogy to many cryptographic protocols based on the discrete logarithm problem. Since these problems work in multiplicative groups F p, but can be also defined over any other group, it was to explore different groups for the purpose of cryptographic systems. There are two major reasons for this. Firstly, other groups may be more efficient to implement in both hardware and software, and secondly, the DLP may be significantly harder to solve over these groups and as a consequence, smaller key sizes will provide the same level of security. There is an enormous amount of groups that can be defined over elliptic curves, which are of very rich structure, providing increased cryptographic strength in comparison to the classical discrete logarithm problem cryptosystems. The strongest known methods for dealing with the discrete log problem do not seem to be applicable to ECs. In particular, in the case of characteristic 2: Known methods for solving the DLP on F 2rmake it possible to break the cryptosystems unless r is chosen to be rather large. Nevertheless, it seems that the analogous systems using ECs over F 2 rwill be secure with significantly smaller values of r. Table 1 shows a comparison between three types of cryptosystems and the necessary key length it requires them to provide the same level of security, i.e. the same amount of work required to break them. One can clearly see that EC security is satisfied with relatively short key lengths. This is a major advantage for embedded systems with limited resources (smartcards, handhelds, cellular phones, etc.). TABLE 1. Cryptographic Strength Comparison Block Cipher Key-length RSA Key Length EC Key Length The security of an ECC, however, depends not only on the length of the key, but also on the elliptic curve parameters. In general, it takes a long time to generate secure parameters.

7 SEMINAR SECURITY - REPORT ELLIPTIC CURVE CRYPTOGRAPHY 7 So, faster parameter generation is important for practical implementation of an ECC. This is a major issue when using elliptic curves or in other words a disadvantage of the rich structure they offer. Choosing unlucky parameters can provide a hook for certain kinds of attacks on such cryptosystems Attacks On EC Cryptosystems. As we have seen above, breaking an EC cryptosystem depends on the difficulty of solving the discrete logarithm problem over elliptic curves. There are two known general methods of solving this problem: (1) The square root method, which is a general method for the discrete logarithm problem. (2) The Silver-Pohlig-Hellman, which factors the order of a curve into small primes and solves the discrete logarithm problem as a combination of discrete logarithms for small numbers. (That s why knowing N and making sure it is not a product of small primes, will boost the confidence in the elliptic curve.) The square root method is the most general attacking method for the discrete logarithm problem, and its computation time is proportional to the exponent of half the key length or in other words, the computation time varies exponentially with respect to the key length. A public key cryptosystem is regarded as being very secure against an attack if the attack takes an exponential amount of time with respect to the key length. From this criterion, we can say that ECCs are very secure against the square root method. The Silver-Pohlig-Hellman method is effective only when the order of the curve is expressed as a product of small primes. Otherwise, the computation time is equivalent to that of the square root method. Therefore, for an ECC, if we select the order of the elliptic curve to be a prime or nearly a prime whose factors include a large prime, the computation time needed to break the ECC will vary exponentially. Therefore a high level of security can be achieved. However, special attacks were found by using the special characteristics of special elliptic curves. The special characteristics are determined by the order of the elliptic curve. These special attacks are much stronger than the square-root method, as for example: The Menezes-Okamoto-Vanstone (MOV) attack using the Weil pairing. The Frey-Rueck attack using the Tate pairing. The attacks on anomalous elliptic curves due to Semaev, Satoh-Araki and Smart. Weil descent (for some special finite fields). Special attacks for special elliptic curves are constantly being discovered, and the security of special curves has been actively discussed in recent years. Especially whether a curve is generated randomly, security should be examined intensively. In addition, one should keep in mind that the evaluation of security changes when a stronger attack is found, because the currently evaluated security level is based on the currently known attacks. Therefore, it is desirable to construct a theoretical evaluation that also considers unknown attacks. Conclusion. We have presented the elliptic curve cryptosystems as a promising alternative to classical public key based cryptosystems. In current applications, they are widely used for key exchange. By giving a brief comparison we pointed out that they are cheaper in resources, at the cost of complexity of the theory. Thus, security is mainly based on this complexity, which controversially also generates insecurity for human intuition. However, attacks up to date have been poor.

8 SEMINAR SECURITY - REPORT ELLIPTIC CURVE CRYPTOGRAPHY 8 REFERENCES [1] Neal Koblitz, A Course in Number Theory and Cryptography, Springer Verlag [2] Neal Koblitz, Algebraic Aspects of Cryptography (Algorithms & Computation in Mathematics), Springer Verlag [3] W. Fulton, Algebric Curves, Benjamin, [4] Neal Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag [5] S. Lang, Elliptic Curves: Diophantine Analysis, Springer-Verlag 1978.