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International Journal of Pure and Applied Mathematics Volume 116 No. 23 2017, 103-107 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu

ON PARAMETERIZATION AND THE POINT OPERATION OF CURVE

1G. Jai Arul Jose, 2Md Mastan, 3Louay A. Hussein Al-Nuaimy 1,2,3 Assistant Professor, Department of Computer Science & MIS, Oman College of Management & Technology, Barka, Sultanate of Oman [email protected], [email protected], [email protected]

Abstract: The curve is also called as the the product of whose distances from two fixed points (- . In it is a a,0) and (a,0), the foci, is 2a units apart and is equal to defined from the given two points which is known as a2.The Cartesian formula of Lemniscates is foci. The foci are at distance 2a from each other. The Figure 1 shows the curve whichx +the y value = curve is defined as the of points in which the sum 2aof a =x 5.− y . of distances to each of two foci is a constant. The curve The polar coordinate of the curve is x = r can also be obtained by the inverse transformation of cos θand y = r sin θ along with the used for inversion centred at the centre of hyperbola. The Cartesian equation of the curve is . This study focuses on the parameterizationx + y = 2a of xthe− curve y and analysis of the point operation to check the suitability for cryptographic operations.

Keywords: Lemniscates, Cryptography, , Parameterization, Elliptic curve.

1. Introduction From ancient time the Algebraic curves have been Figure 1. Lemniscates Curve studied. We, all are very familiar with Circle, and , those are examples of conic sections. Ancient 2. Parameterization of Leminiscates Greeks have studied different types of curves. The Let we set . (Castillon, 1741), the lemniscate (Jacob Then thet = equation x + y of Lemniscate will become Bernoulli, 1694) and the folium (Descartes, 1638) are recent curves which are more familiar. All of these t = 2a x − y = 2a x + y − 2y curves stake the property which, at the side of their = 2a t − 2y i.e.; ⇒ ⇒ geometrical description, they could be given by algebraic equalities in the plane fitted out with coordinates x andy. t − 2a t = −4a y y = y = ± In 1964 James Bernoulli, an individual from eminent √ Bernoulli family of mathematicians published his also, we can write the equation of the curve as solutions on a curve that he called it as lemniscus. The t = 2a x − y = 2a 2x − x − y Latin word lemniscusmeans ribbon. This lemniscus curve = 2a 2x − t is the special category of a Cassinian Oval and the i.e.; ⇒ ⇒ arclength of this curveturn into very significant for future t + 2a t = 4a x x = x = research on elliptical functions. The Lemniscates curve is ± √ symmetric about the origin, the x-axis and the y-axis; Here there are two issues; the first one is how to which is an important property of this curve. The choose the sign of x and y; the second one at which definition of Lemniscates is the locus of a point, which interval the t should lie on.

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For the first issue, if we have a point (x, y) on the The Lemniscates might appear uncharacteristic at initial curve, the remaining points are (x, -y), (-x, y) and (-x, - look, but various parallels exist amongst it and of the sine y). So it is enough to parameterize the curve on the first function. For example, the sine function may be defined quadrant. The rest of the curve can be determined by the as the inverse integral function as follows: symmetry property of the curve. -1 y = sin s ⇔s = sin y= . dt Therefore, we take, √ The Lemniscates function√ ϒ = ϕ(s) can also be x = defined as inverse function of an integral and . ϒ √ ϒ = ϕ(s) ⇔s= . For= the second issue, we note that the curve crosses dt the x-axis at (1, 0) and passes through the origin. The √ point (0, 0) links to t = 0 and (1, 0) links to t = 1. So C. The characteristics of ϕϕϕ(s) tshould be in the interval [0, 1] The Lemniscates function satisfies numerous interesting identities: 3. The and Lemniscates Function Proposition 1: A. The Arc Length of Lemniscates If f(x) = sin x, then: 1) f(x+2 π) = f(x) Let us consider a> 0 where a is a real number. 2) f(-x) = -f(x) Let F 1and F 2be the foci (a, 0) and (-a, 0) respectively 2 3) f( π - x) = f(x) on R . /2 2 4) f (x) = 1 – f (x) Let C = { P ∈R2; PF . PF = a 2 }. 1 2 The Lemniscates function ϕ(s) fulfils similar Let us derive the equation of the curve C on polar identities. In fact, we may view the Lemniscates function coordinates. as a speculation of the sine function for various curve. Of We know that P = (r cos θ, r sin θ): course, the sine function is only significant with respect Then θ with respect to the unit circle, whereas ϕ(s) pertains to θ PF = r + a − 2ar cos , PF = r + a + the Lemniscates. We observe that the below propositions 2arThus, cos θ θ are true of the Lemniscates function: r + a − 2arθ cos r + a + 2ar cos = Proposition 2: r + a − 4a r cos = a θ If f(s) = ϕ(s), then: r + 2r a + a −θ 4a r cos = aθ st 1) f(s+2 ω) = f(s) In caser of= 2a P ∈C2 is cos on the− 1 1 = quadrant. 2a cos 2 Let sbe the 2) f(-s) = -f(s) length of arc among O = (0, 0) &P. 3) f( ω - s) = f(s) θ θ θ 4) f/2(s) = 1 – f4(s) Therefore θ . Since θ , The identities 1, 2 and 3 are easy to observe. The 4th = r + d d = dr θ θ of Proposition 1 is simply rewritten of the well-known . 2 2 identity cos x = 1 – sin x, where cos x is, in fact, the s = r + dr θ differentiation of sin x. Now though the resemblance Hence . amongst this identity and the equivalent identity for the = 1 + r dr θ θ Lemniscates function is clear, this is the least intuitive As θ θ θ identity of ϕ(s). θ r = a√2θ cos 2 , =θ − √ , Thus = √ θ θ − that is = 4. The Point Operation of Lemniscates θ θ θ r The point operations are described in terms of the cos 2 = , sin 2 = 1 − cos 2 θ Lemniscates function. 2a 4a − r d r = Hence = Therefore the arc length4a of Lemniscatedr is 4a − r A. The Addition and Subtraction laws for ϕϕϕ(s)

The sine trigonometry function satisfies the law of 4a 2a addition sin(x+y) = sin x cos y + cos x sin y. So, if we / / s = dr = dr say f(x) = sin(x), then f(x+y) = f(x)f (y) + f (x)f(y). We 4a − r √4a − r derive a related result for ϕ(s), starting with the below B. The Lemniscates Function identity:

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α β Lemniscates with respect to the arc length s could be 1 1 constructed by straightedge and compass iff ϒ=ϕ(s) is a dt + dt constructible number. By Noting that as the Lemniscates 1 − t ϒ 1 − t be defined by the equation 1 and that ϒ2 = x 2+y 2, we note thatx + ϒ y4 = x =2 – 2a y2. xThen−y by = dt 1 − t solving in terms of ϒ, we see that: α β β α where α, β∈ [0, 1] and ϒ √ ∈ [0, 1] α β = ± ± By allowingx, y and z equal the 3 integrals above, 1 1 x = r + r ; y = r − r respectively, and applying the ϕ function to both of the 2 2 sides of the equation, we get 5. The Cryptographic Scheme on Lemniscates curve α β β α ω with to ECC ϕ(x+y) = ϕ(z) = ϒ √ , 0 ≤ x+y ≤ . α β = The discussion in previous section are by considering the Now, since ϕ(x) = α and ϕ(y) = β, we have numbers on the set of all real numbers. Usually real ϕ ϕ ϕ ϕ ϕ(x+y) = ϕ(z) = ϒ , 0 ≤ number operations comparatively slow with integer ϕ ϕ ω = number operations. Also sometimes the result will not be x+y ≤ . accurate due to the round-off error in Real numbers. And the last of our basic ϕ properties implies that Operations in Cryptographic scheme should be accurate ϕ = ϕ′ , yielding and of course faster. To achieve this, we need to consider 1 − x x the points of the curve over the prime field F P. We select ϕ ϕ′ ϕ′ ϕ ω the field in such a way that it should finitely large ϕ , 0 ≤ x+y ≤ . ϕ ϕ number of points suitable for cryptographic operations. x + y = Now since both of the sides of this equation are analytic functions of x that are defined ∀x when y is any A. Lemniscates on Prime field F p fixed value, the equation holds true ∀x and y. The equation of Lemniscates on a prime field F is The subtraction law for ϕ(s) can be easily derived p ≡ from the addition law. Since ϕ(-x) = -ϕ(x) and ϕ/(-x)= / x Here+ y amodp 2a ≠x 0 and− y p ismod a prime p number. ϕ (x) Here the members of the field are integers from0top – ϕ ϕ′ ϕ′ ϕ ϕ 1. All arithmetic operations involve whole numbers x yϕ − ϕx y between 0 and p – 1. In order to make the security of this x − y = 1 + x y scheme to be stronger, we should choose the prime B. Scalar Multiplication number in such a way that there exists a finitely large number of points on Lemniscates. The algebraic rules for From the addition law, we can get easily ϕ(2x) = addition, subtraction and doubling of points discussed in ϕ ϕ | . ϕ the previous section can be used over the prime field Fp. Then by replacing x and y with 2x and x, respectively, we have B. Domain parameter over F p ϕ ϕ′ ϕ(3x)+ ϕ(x) = ϕ(2x+x)+ ϕ(2x-x) = . The Fpdomain parameters are a 5-tuple T=(p,a,G,n,h); ϕ ϕ where Now using the doubling formula p– the prime number for the finite field F p, ϕ ϕ′ ϕ(2x) = a– the parameter of the curve, ϕ G– the generator point (x G,y G), a point on this curve ϕ ϕ | ϕ which have been chosen for Cryptographic operations, ϕ | 2 x n– the order of the curve. The scalar for point ϕ ϕ ϕ ϕ multiplication is chosen from a number between 0 and n 3x + x = | ϕ ϕ – 1, 1 + /2 x4 And finally, since ϕ(x) = 1 - ϕ (x), we have our h– the cofactor where h = #F(F p)/n.#F(F p), is the total result: number of points taken on the Lemniscates curve. ϕ ϕ ϕ(3x) = ϕ(x) . C. The Key Pairs ϕ ϕ Now, with this understanding, let us explore the construction on the Lemniscates. The point over the

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In public key cryptosystem for secure transmission [4] V. Miller, “Uses of Elliptic Curves in key exchange is used without considering any third Cryptography”, Advances in Cryptology; proceedings of parties. The key exchange scheme is described here.For a Cryto’85, pp. 417-426,1986. chosen Lemniscates domain parameters T=(p,a,G,n,h), or [5] Benbouziane T., Houary, H., Kahoui, M.: Polynomial T=(m,f(x),a,G,n,h), a Lemniscates curve key pair (d,Q) Parametrization of nonsingular algebraic curves. related with Tcontainsa secrete key d, which is taken University of Firenze, Firenze, 2000.AMenezes, P. Van from the interval [1,n – 1], and a public key Q=(x Q, y Q) OOrschot, and S.Vanstone. Handbook of Applied which is actually the point Q=dG. Cryptography. CRC press, 1997. [6] Gahleitner, M., J¨uttler, B., Schicho, J.: D. Encryption and Decryption Approximate parametrization of planar cubics. InCurve and Surface Fitting. Nashboro Press, St. Malo, 2002. To send an encrypted message P to a receiver, say B, m [7] Certicom Research. SEC 2:Recommended Elliptic the sender, say A, select a random positive whole number Curve Domain parameters. Standards or efficient k and results the cipher text C containing the pair of m Cryptography Version 1.0, Sep 2000. points. [8] C. Sajeev, G. Jai Arul Jose, “Elliptic Curve C =[kG, P +kP ] m m B Cryptography Enabled Security for Wireless Here, A used B’s public key P . To get back the B Communication”, International Journal on Computer original plain text from the cipher text, B find the product Science and Engineering, Vol. 02, No. 06, pp. 2187- of the first point in the pair and B’s private key n then B 2189, 2010. subtracts the result from the second point as given below: [9] Neal Koblitz. 2007. The uneasy relationship P +kP –n (kG)=P +k(n G) – n (kG)=P m B B m B B m between Mathematics and Cryptography. Notes of the Key sharing between the users A&B can be dond as AMS, 54(8):972-979. follows: [10] P´erez-D´ıaz, S.: On the problem of proper 1. A select n , an integer

with a rational pencil Proceedings of the 2000 6. Conclusion international symposium on Symbolic and algebraic The Lemniscates has many very interesting properties computation, pp.292-300. St. Andrews, 2000. over it. One of the important properties of the [13] Junfeng Fan, Kazuo Sakiyama, Ingrid Lemniscates curve is its symmetricity. The addition Verbauwhede. 2008. Elliptic curve cryptography on operation is very compatible with additive group over F p. embedded multicore systems. Journal of Design The point operations are described in the previous Automation for Embedded Systems, 123-134. sections. The ECC methodology is used for key [14] Kanniah, Samsudin. 2007. Multi-threading elliptic generation and exchange also for encryption. Time curve cryptosystems. Proceedings of complexity has to be analysed in future work. As the Telecommunications and Malaysia International parametric form and the point operations are well conference on communications, 134-139. defined, Lemniscates can be used for cryptographic key [15] Lee, Wong. 2004. A random number generator pairing and other operations. based elliptic curve operations. Computers and Mathematics with Applications, 47:217-226. Reference [1] Cox, David A. Galois Theory. Hoboken, NJ: John Wiley & Sons, 2004. pp. 457-508. [2] N. Koblitz, “Elliptic Curve Cryptosystems”, Mathematics of Computation, vol. 48, pp. 203-209. 1987. [3] Bastl, B.: Aplikacegeometrie II. Pomocn´yuˇcebn´ı text, Z´apadoˇcesk´auniverzita, Plzeˇn, 2005.

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