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(IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 8, August 2012 Elimination of Weak Elliptic Curve Using Order of Points Nishant Sinha#1, Aakash Bansal*2 # School of IT CDAC Noida, India 1 sinha22nishant@gmail.com * School of IT CDAC Noida, India 2 aakashbansal.cdac@gmail.com Abstract-The elliptic curve cryptography (ECC) is a public Only the particular user knows the private key where as key cryptography. The mathematical operations of ECC is the public key is distributed to all users taking part in the defined over the elliptic curve y2=x3+ax+b, where communication. Public key cryptography, unlike private 4a3+27b2ǂ0. Each value of the ‘a’ and ‘b’ gives a different key cryptography does not require any shared secret elliptic curve. All points (x,y) which satisfies the above between communicating parties but it is much slower than equation plus a point at infinity lies on the elliptic curve. private key cryptography which is main drawbacks of There are certain property of elliptic curve which makes the cryptography weak. In this paper, we have proposed public key cryptography. technique which would eliminate such weak property and Elliptic curve cryptography is a variant of public key will make elliptic curve cryptography more secure. cryptography which eliminates the drawback of public cryptography. Elliptic curve y2=x3+ax+b, where 4a3+27b2 Keywords: cryptography, security, anomalous curve, discrete ǂ0 for which each value of ‘a’ and ‘b’ gives a different logarithm problem elliptic curve. In ECC, public key is the point on the curve I INTRODUCTION and private key is a random number. The public key is obtained by multiplying the private key with the generator Cryptography is the study of “mathematical” systems for point G in the curve. solving two kinds of security problems: privacy and One main advantage of ECC is its small size. A 160 bit authentication [1].Two types of cryptography are present key in ECC is considered to be as secured as 1024 bit key – private key cryptography and public key cryptography. in RSA. In public key cryptography, each user or the device taking II BACKGROUND KNOWLEDGE part in the communication generally have a pairs of keys, a public key and a private key, and a set of operations Elliptic Curves associated with the key to do the cryptographic Elliptic curves are not ellipses, instead, they are cubic operations. curves of the form y2 = x3 + ax + b. Elliptic curves over 48 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 8, August 2012 R2 (R2 is the set R x R, where R = set of real numbers) is point O, which is the point at infinity and which is the defined by the set of points (x, y) which satisfy the identity element under addition. 2 3 equation y = x + ax + b, along with a point O, which is Similar to E(Fp), addition is defined over E(F2m) and we the point at infinity and which is the additive identity can similarly verify that even E(F2m) forms an abelian element. The curve is represented as E(R). group under addition. The following figure is an elliptic curve satisfying the equation y2 = x3 – 3x + 3 :- B. Advantage of Elliptic Curve Cryptography Over RSA/DSA The advantage of elliptic curve over the other public key systems such as RSA, DSA etc is the key strength[2]. The following table summarizes the key strength of ECC based systems in comparison to other public key schemes. RSA/DSA Key ECC Key Length for Equivalent length Security 1024 160 2048 224 Elliptic curve over R2: y2 = x3 – 3x + 3 3072 256 A. Elliptic Curves over Finite Fields 7680 384 1) Elliptic Curves over Fp: An elliptic curve E(Fp) over a finite field Fp is defined by the parameters a, b ∈ Fp (a, b 15360 512 satisfy the relation 4a3 + 27b2 ≠ 0), consists of the set of points (x, y) ∈ Fp, satisfying the equation y2 = x3 + ax + b. Table 1:-Comparison of the key strengths of RSA/DSA and ECC The set of points on E(Fp) also include point O, which is From the table it is very clear that elliptic curves offer a the point at infinity and which is the identity element comparable amount of security offered by the other under addition. popular public key for a much smaller key strength. This 2) Elliptic curves over F2m:An elliptic curve E(F2m) over a property of ECC has made the scheme quite popular of finite field F2m, is defined by the parameters a, b ∈ F2m, late. (a, b satisfy the relation 4a3 + 27b2 ≠ 0, b ≠ 0), consists of the set of points (x, y) ∈ F2m, satisfying the equation y2 + xy = x3 + ax + b. The set of points on E(F2m) also include 49 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 8, August 2012 III ELLIPTIC CURVE DISCRETE LOGARITHM over the finite field Fq with q = pⁿ , n ∈ Z+ and p a The strength of the Elliptic Curve Cryptography lies in prime. Then there exists a unique t ∈ Z such that the Elliptic Curve Discrete Log Problem (ECDLP). The #E(Fq) = q + 1 - t where |t| < 2√q.[4] statement of ECDLP is as follows. B. Reducing the problem of computing the order of curve Let E be an elliptic curve and P ∈ E be a point of order n. #E(Fpn) to #E(Fp) Given a point Q ∈ E with Q = mP, for a certain m ∈ {2, It tells that if we can compute #E(Fp), then we can 3, ……, m – 2}. compute #E(F pⁿ) in a direct manner.Let #E(Fp) = p + 1 Find the m for which the above equation holds. - t. When E and P are properly chosen, the ECDLP is thought Write X2 - t X + p = (X – α) (X – β). to be infeasible. Note that m = 0, 1 and m – 1, Q takes the Then αⁿ +βⁿ ∈ Z and #E(F pⁿ) = pⁿ + 1 –(αⁿ +βⁿ) . values O, P and – P. One of the conditions is that the order of P i.e. n be large so that it is infeasible to check all If p is a small prime, then it is easy to determine #E(Fp) the possibilities of m. by direct counting or other simple methods. The difference between ECDLP and the Discrete C. Weak curves Logarithm Problem (DLP) is that, DLP though a hard 1) Anomalous curve: The curve E(Fq) is said to be problem is known to have a sub exponential time anomalous if # E(Fq) = q. These curves are weak when solution, and the solution of the DLP can be computed q=p, the field characteristic. faster than that to the ECDLP. This property of Elliptic curves makes it favorable for its use in cryptography. 2) Supersingular elliptic curves: The MOV(Menezes, A direct approach to determining # E(Fq) is to compute z Okamoto, and Vanstone) attack on elliptic curves = x3 + A x + B for each x ∈ Fq, and then to test if z has a shows that ECDLP can be reduced to the classical square root in Fq. If z = 0, then (x, 0) ∈ E(Fq). discrete logarithm problem on some extension field Fqk , for some integer k (k is called the embedding If there exists y ∈ Fq such that y2 mod q= z, then (x,y),(x,- degree or MOV degree). The MOV attack is only y) ∈ E(Fq) , else there is no point in E(Fq)with x- practical when k is small. For Supersingular elliptic coordinate x. So there are at most 2 q + 1 elements in the curves k<=6. group. 3) Prime-field anomalous curves: If #E(Fp) = p, there is A theorem of finite fields states that exactly 1/2 of the polynomial algorithm solving the ECDLP by lifting the non-zero elements of Fq are quadratic residues. So on curve and points to Z. average, there will be approximately q + 1 elements in E(Fq). The given properties of weak curve indicate that the order of elliptic curve plays a major role in determining whether A. Hasse's Theorem the given curve is weak or not. The Prime-field anomalous curve and anomalous curve where the order of The following theorem, first proved by Helmut Hasse, told bounds on # E(Fq) . Let # E(Fq) be an elliptic curve 50 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 8, August 2012 curve is a prime number can be identified with the help of that if the value of x1 is put in the equation x3 + ax + b Lagrange’s Theorem and Hasse’s Theorem. then it will be equal to zero. Because of these reasons, in step 1 of the algorithm the solution of equation x3 + ax + b = 0 is determined and IV PROPOSED APPROACH check wether the solution lies in the field in which elliptic A. Lagrange’s Theorem curve is defined. If G is a finite group and H is a subgroup of G, then |H| D. Facts derived from above algorithm divides |G| i.e. order of subgroup H will divides the order of group G and the order of each element of the group 1) The set of points E(Fq) is a finite abelian group. It is divides the order of the group [5]. always cyclic or the product of two cyclic groups. For By using the above theorem an algorithm is developed to example the curve defined by over examine that the curve may have the property of F71 has 72 points (71 affine points including (0,0) and one Anomalous curve and Prime-field anomalous curve. point at infinity) over this field, whose group structure is given by Z/2Z × Z/36Z. B. Proposed Algorithm If the order of elliptic curve is prime then 3 Step 1:- Find the solution of Equation x + ax + b=0 according to fundamental theorem of finite abelian group which is the right hand side portion of general elliptic it is isomorphic to Zn where n is prime and it is always cuve equation y2 = x3 + ax + b. cyclic group. Step 2:- Determine whether the solution of the above 2) If the order of elliptic curve is prime then every point of equation lies in the field where elliptic curve equation is elliptic curve can play the role of generator in elliptic defined. curve cryptography. Step 3:- If the solution exist in the the field then there is atleast a point (x1, y1) of order two i.e. 2(x1, y1)=0 which 3) The elliptic curve which has points of order 2 signifies indicate that order of the elliptic curve can not be a prime that the order of elliptic curve is even number which number. reduces the range of Hasse’s bound theorem which tells that order of the elliptic curve #E(Fq) = q + 1 - t where C. Correctness of above algorithm |t| < 2√q . If there is a point (x1, y1) of order two lies on the elliptic curve, then (x1, y1) + (x1, y1) = 0 which is point at infinity.This implifies that (x1, y1) = - (x1, y1). V CONCLUSION From the arithmetic of elliptic curve, it is known that - For efficient implementation of ECC, it is important that (x1, y1) is a point which is mirror image of (x1, y1) with there must be some constraints on order of the elliptic respect to X-axis. So (x1, y1) = - (x1, y1) is true only when curve. In our study, we have found that there are some the Y-coordinates of (x1, y1) is equal to zero. It indicate curves which are not suitable for elliptic curve 51 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 8, August 2012 cryptography because of their weak properties. These weak properties are based on the order of the elliptic curve. We have developed procedure which can identify prime-field anomalous curves which is weak and not suitable for cryptography .The proposed procedure also reduces the range of order of the elliptic curve by half. ACKNOWLEDGMENTS The authors would like to thank the anonymous reviewers for the valuable comments that have significantly improved the paper quality. They would also like to thanks their respective head of departments for the selfless guidance which encourage them to do this research. REFERENCES [1] William Stallings, Cryptography and Network Security-Principles and Practice, Prentice Hall Publications, Second Edition. [2] A. K Lenstra, E.R.Verhul, “Selecting Cryptographic key sizes”, Nov 14 1999. [3] Ian F. Blake, Gadiel Seroussi, and Nigel P. Smart, Elliptic Curves in Cryptography, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1999 [4] Advances in Elliptic Curve Cryptography (Edited by I.F. Blake, G. Seroussi and N.P. Smart). London Mathematical Society Lecture Note Series, Cambridge University Press, 2004. [5] A Menezes, S. Vanstone, T. Okamoto, ”Reducing Elliptic Curve Logarithms to Logarithms in a Finite Field”, IEEE transaction on Information Theory, Vol 39 (1993), 1639-1646. [6] B.Schneier ,Applied Cryptography ,John Wiley and Sons, Second Edition, 1996. [7] Alessandro Cilardo, Luigi Romano, Nicola Mazzocca and Luigi Coppolino, “Elliptic Curve Cryptography Engineering” PROCEEDINGS OF THE IEEE, VOL. 94, NO. 2, FEBRUARY 2006. [8] Lawrence C. Washington , Elliptic Curves: Number Theory and Cryptography, 2nd edition . 52 http://sites.google.com/site/ijcsis/ ISSN 1947-5500

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