On Parameterization and the Point Operation of

Book of Proceedings : ICCAMEET2017 International Conference on Contemporary Approaches In Mathematics And Emerging Engineering Trends

1

On Parameterization and the Point Operation of Lemniscates Curve G. Jai Arul Jose1, Md Mastan2, Louay A. Hussein Al-Nuaimy3 1,2,3

Assistant Professor, Department of Computer Science & MIS, Oman College of Management & Technology, Barka, Sultanate of Oman 1 [email protected] 2 [email protected] 3 [email protected] Abstract±The Lemniscate curve is also called as the Lemniscate of Bernoulli. In geometry it is a plane curve defined from the given two points which is known as foci. The foci are at distance 2a from each other. The curve is defined as the locus of points in which the sum of distances to each of two foci is a constant. The curve can also be obtained by the inverse transformation of hyperbola along with the circle used for inversion centred at the centre of hyperbola.The Cartesian equation of the curve isሺ‫ ݔ‬ଶ ൅ ‫ ݕ‬ଶ ሻଶ ൌ ʹܽଶ ሺ‫ ݔ‬ଶ െ ‫ ݕ‬ଶ ሻ. This study focuses on the parameterization of the curve and analysis of the point operation to check the suitability for cryptographic operations. Keywords±Lemniscates, Cryptography,Algebraic Curve, Parameterization, Elliptic curve.

I. INTRODUCTION From ancient time the Algebraic curves have been studied. We, all are very familiar with Circle, Ellipse and Parabola, those are examples of conic sections. Ancient Greeks have studied different types of curves. The cardioid (Castillon, 1741), the lemniscate (Jacob Bernoulli, 1694) and the folium (Descartes, 1638) are recent curves which are more familiar. All of these curves stake the property which, at the side of their geometrical description, they could be given by algebraic equalities in the plane fitted out with coordinates x andy. In 1964 James Bernoulli, an individual from eminent Bernoulli family of mathematicians published his solutions on a curve that he called it as lemniscus. The Latin word lemniscusmeans ribbon. This lemniscus curve is the special category of a Cassinian Oval and the arclength of this curveturn into very significant for future research on elliptical functions. The Lemniscates curve is symmetric about the origin, the x-axis and the y-axis; which is an important property of this curve. The definition of Lemniscates is the locus of a point, which the product of whose distances from two fixed points (a,0) and (a,0), the foci, is 2a units apart and is equal to a2.The Cartesian formula of Lemniscates isሺ‫ ݔ‬ଶ ൅ ‫ ݕ‬ଶ ሻଶ ൌ ʹܽଶ ሺ‫ ݔ‬ଶ െ ‫ ݕ‬ଶ ሻǤFigure 1 shows the curve which the value of a = 5. The polar coordinate equation of the curve is x = r cosTand y = r sin T

Fig. 1 Lemniscates Curve

II.PARAMETERIZATION OF LEMNISCATES Let we set ‫ ݐ‬ଶ ൌ ‫ ݔ‬ଶ ൅ ‫ ݕ‬ଶ . Then the equation of Lemniscate will become ‫ ݐ‬ସ ൌ ʹܽଶ ሺ‫ ݔ‬ଶ െ ‫ ݕ‬ଶ ሻ ൌ ʹܽଶ ሺ‫ ݔ‬ଶ ൅ ‫ ݕ‬ଶ െ ʹ‫ ݕ‬ଶ ሻ ൌ ʹܽଶ ሺ‫ ݐ‬ଶ െ ʹ‫ ݕ‬ଶ ሻ ସ ଶ ଶ i.e.;‫ ݐ‬െ ʹܽ ‫ ݐ‬ൌ െͶܽଶ ‫ ݕ‬ଶ Ÿ‫ ݕ‬ଶ ൌ ଶ௔మ ௧ మ ି௧ ర

ඥଶ௔మ ௧ మ ି௧ ర

Ÿ‫ ݕ‬ൌ r ଶ௔ also, we can write the equation of the curve as ‫ ݐ‬ସ ൌ ʹܽଶ ሺ‫ ݔ‬ଶ െ ‫ ݕ‬ଶ ሻ ൌ ʹܽଶ ሺʹ‫ ݔ‬ଶ െ ‫ ݔ‬ଶ െ ‫ ݕ‬ଶ ሻ ൌ ʹܽଶ ሺʹ‫ ݔ‬ଶ െ ‫ ݐ‬ଶ ሻ ସ௔మ

r

i.e.;‫ ݐ‬ସ ൅ ʹܽଶ ‫ ݐ‬ଶ ൌ Ͷܽଶ ‫ ݔ‬ଶ Ÿ‫ ݔ‬ଶ ൌ

ඥଶ௔మ ௧ మ ା௧ ర ଶ௔

ଶ௔మ ௧ మ ା௧ ర ସ௔మ

Ÿ‫ ݔ‬ൌ

Here there are two issues; the first one is how to choose the sign of x and y; the second one at which interval the t should lie on. For the first issue, if we have a point (x, y) on the curve, the remaining points are (x, -y), (-x, y) and (-x, -y). So it is enough to parameterize the curve on the first quadrant. The rest of the curve can be determined by the symmetry property of the curve. Therefore, we take, ‫ ݔ‬ൌ ඥଶ௔మ ௧ మି௧ ర

ඥଶ௔మ ௧ మ ା௧ ర ଶ௔

and ൌ . ଶ௔ For the second issue, we note that the curve crosses 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 tshould be in the interval [0, 1]

ISBN: 978-93-81899-77-9

Book of Proceedings : ICCAMEET2017 International Conference on Contemporary Approaches In Mathematics And Emerging Engineering Trends

2 III. THE ARC LENGTH AND LEMNISCATES FUNCTION A. The arc length of Lemniscates Let us consider a> 0 where a is a real number. Let F1and F2be the foci (a, 0) and (-a, 0) respectively on R2. Let C = { PR2; PF1 . PF2= a2 }. Let us derive the equation of the curve C on polar coordinates. We know that P = (r cos T, r sin T): Then ܲ‫ܨ‬ଵଶ ൌ ‫ ݎ‬ଶ ൅ ܽଶ െ ʹܽ‫ •‘… ݎ‬T ǡ ܲ‫ܨ‬ଶଶ ൌ ‫ ݎ‬ଶ ൅ ܽଶ ൅ ʹܽ‫ •‘… ݎ‬T ሺ‫ ݎ‬ଶ ൅ ܽଶ െ ʹܽ‫ •‘… ݎ‬Tሻሺ‫ ݎ‬ଶ ൅ ܽଶ ൅ Thus, ଶ ʹܽ‫ •‘… ݎ‬Tሻ ൌ ሺ‫ ݎ‬൅ ܽଶ ሻଶ െ Ͷܽଶ ‫ ݎ‬ଶ …‘• ଶ T ൌ ܽସ ‫ ݎ‬ସ ൅ ʹ‫ ݎ‬ଶ ܽଶ ൅ ܽସ െ Ͷܽଶ ‫ ݎ‬ଶ …‘• ଶ T ൌ ܽସ ‫ ݎ‬ଶ ൌ ʹܽଶ ሺʹ …‘• ଶ T െ ͳሻ ൌ ʹܽଶ …‘• ʹT In case of PC is on the 1st quadrant. Let sbe the length of arc among O = (0, 0) &P. ଶ

T ௗT Therefore ൌ ‫׬‬଴ ට‫ ݎ‬ଶ ൅ ቀ ቁ ݀T . Since

݀T ൌ

ௗT

ௗ௥

ௗ௥

௥ ௗT ݀‫ݎ‬, ‫ ݏ‬ൌ ‫׬‬଴ ට‫ ݎ‬ଶ ൅ ቀ ቁ ௗ௥ ଶ

ିଶ ௗT ௗ௥

௥ ௗT Hence ൌ ‫׬‬଴ ටͳ ൅ ‫ ݎ‬ଶ ቀ ቁ ݀‫ ݎ‬. ௗ௥

݀‫ݎ‬.

As ௗ௥ ଶ௔ ୱ୧୬ ଶT ௗT ‫ ݎ‬ൌ ܽξʹ …‘• ʹT ǡ ൌ െ ǡ ݄ܶ‫ ݏݑ‬ൌ ξଶ ୡ୭ୱ ଶT

ௗT

ௗT ଶ

ξଶ ୡ୭ୱ ଶT ୡ୭ୱ ଶT

ௗ௥

‫ ݏ݅ݐ݄ܽݐ‬ቀ ቁ ൌ మ మ ௗ௥ ଶ௔ ୱ୧୬ ଶT ‫ݎ‬ଶ …‘• ʹT ൌ ଶ ǡ •‹ଶ ʹT ൌ ͳ െ …‘• ଶ ʹT ʹܽ Ͷܽସ െ ‫ ݎ‬ସ ݀T ଶ ൌ ‫݁ܿ݊݁ܪ‬ ൬ ൰ ݀‫ݎ‬ Ͷܽସ ଶ ‫ݎ‬ ൌ ସ Ͷܽ െ ‫ ݎ‬ସ Therefore the arc length of Lemniscate is െ

ଶ௔ ୱ୧୬ ଶT

௥

௥

ʹܽଶ Ͷܽସ ݀‫ݎ‬ ݀‫ݎ‬ ൌ න ‫ ݏ‬ൌනඨ ସ Ͷܽ െ ‫ ݎ‬ସ ξͶܽସ െ ‫ ݎ‬ସ ଴

଴

B. The Lemniscates Function The Lemniscates might appear uncharacteristic at initial look, but various parallels exist amongst it and of the sine function. For example, the sine function may be defined as the inverse integral function as follows: ௬ ଵ y = sin sœs = sin-1y= ‫׬‬଴ ݀‫ݐ‬. మ ඥଵି௧

The Lemniscates function b = M(s) can also be defined as inverse function of an integral b ଵ ݀‫ ݐ‬. b = M(s) œs= ‫׬‬଴ మ ඥଵି௧

C. The characteristics of M(s) The Lemniscates function satisfies numerous interesting identities: Proposition 1: If f(x) = sin x, then:

1) f(x+2S) = f(x) 2) f(-x) = -f(x) 3) f(S - x) = f(x) 4) f/2(x) = 1 ± f2(x) The Lemniscates function M(s) fulfils similar identities. In fact, we may view the Lemniscates function as a speculation of the sine function for various curve. Of course, the sine function is only significant with respect with respect to the unit circle, whereas M(s) pertains to the Lemniscates. We observe that the below propositionsare true of the Lemniscates function: Proposition 2: If f(s) = M(s), then: 1) f(s+2Z) = f(s) 2) f(-s) = -f(s) 3) f(Z - s) = f(s) 4) f/2(s) = 1 ± f4(s) The identities 1, 2 and 3 are easy to observe. The 4th of Proposition 1 is simply rewritten of the well-known identity cos2x = 1 ± sin2x, where cos x is, in fact, the differentiation of sin x. Now though the resemblanceamongst this identity and the equivalent identity for the Lemniscates function is clear, this is the least intuitive identity of M(s). IV. THE POINT OPERATION OF LEMNISCATES The point operations are described in terms of the Lemniscates function. A. The Addition and Subtraction laws for M(s) The sine trigonometry function satisfies the law of addition sin(x+y) = sin x cos y + cos x sin y. So, if we say f(x) = sin(x), then f(x+y) = f(x)f/(y) + f/(x)f(y). We derive a related result for M(s), starting with the below identity: D E ͳ ͳ ݀‫ ݐ‬൅ න ݀‫ݐ‬ න ସ ଴ ඥሺͳ െ ‫ ݐ‬ሻ ଴ ඥሺͳ െ ‫ ݐ‬ସ ሻ b ͳ ݀‫ݐ‬ ൌන ଴ ඥሺͳ െ ‫ ݐ‬ସ ሻ 1]

where D, E [0, 1] and b ൌ

DටଵିEర ାEඥଵିDర ଵାDమ Eమ

 [0,

By allowingx, y and z equal the 3 integrals above, respectively, and applying the M function to both of the sides of the equation, we get

Z ଶ

.

M(x+y) = M(z) = b ൌ

DටଵିEరାEඥଵିDర ଵାDమ Eమ

” x+y”

Now, since M(x) = D and M(y) = E, we have

M(x+y) = M(z) = b ൌ Z

Mሺ௫ሻඥଵିMర ሺ௬ሻାMሺ௬ሻඥଵିMర ሺ௫ሻ ଵାMమ ሺ௫ሻMమ ሺ௬ሻ

,

”x+y” . ଶ And the last of our basic M properties implies that ඥͳ െ Mସ ሺ‫ݔ‬ሻ= Mᇱ ሺ‫ݔ‬ሻ, yielding

ISBN: 978-93-81899-77-9

Book of Proceedings : ICCAMEET2017 International Conference on Contemporary Approaches In Mathematics And Emerging Engineering Trends

3 Mሺ‫ ݔ‬൅ ‫ݕ‬ሻ ൌ

Mሺ୶ሻMƍ ሺ୷ሻାMƍ ሺ୶ሻMሺ୷ሻ

Z

The equation of Lemniscates on a prime field Fp

ଶ

is ሺ‫ ݔ‬ଶ ൅ ‫ ݕ‬ଶ ሻଶ {ʹܽଶ ሺ‫ ݔ‬ଶ െ ‫ ݕ‬ଶ ሻሺ݉‫݌݀݋‬ሻ Here amodp  and p is a prime number. Here the members of the field are integers from0top ± 1. All arithmetic operations involve whole numbers between 0 and p ± 1. In order to make the security of this scheme to be stronger, we should choose the prime number in such a way that there exists a finitely large number of points on Lemniscates. The algebraic rules for addition, subtraction and doubling of points discussed in the previous section can be used over the prime field Fp.

”x+y” .

ଵାMమ ሺ୶ሻMమ ሺ୷ሻ

Now since both of the sides of this equation are analytic functions of x that are defined x when y is any fixed value, the equation holds true x and y. The subtraction law for M(s) can be easily derived from the addition law. Since M(-x) = -M(x) and M/(-x)= M/(x) MሺšሻMƍ ሺ›ሻ െ Mƍ ሺšሻMሺ›ሻ Mሺ‫ ݔ‬െ ‫ݕ‬ሻ ൌ ͳ ൅ Mଶ ሺšሻMଶ ሺ›ሻ B. Scalar Multiplication

From the addition law, we can get easily M(2x)

=

ଶMሺ௫ሻMȁ ሺ௫ሻ ଵାMర ሺ௫ሻ

.

Then by replacing x and y with 2x and x, respectively, we have

M(3x)+ M(x) = M(2x+x)+M(2x-x) = Now using the doubling formula

M(2x) =

ଶMሺ୶ሻMƍ ሺ୶ሻ ଵାMర ሺ୶ሻ

Mሺ͵‫ݔ‬ሻ ൅ Mሺ‫ݔ‬ሻ ൌ

൭ʹ

ቀଶMሺ௫ሻMȁ ሺ௫ሻቁ

ͳ൅ቀ

ଵାMర ሺ௫ሻ

ଶMሺ௫ሻMȁ ሺ௫ሻ ଵାMర ሺ௫ሻ

ଶMሺଶ୶ሻMƍ ሺ୶ሻ

ଵାMమ ሺଶ୶ሻMమ ሺ୶ሻ

.

Mȁ ሺ‫ݔ‬ሻ൱ ଶ

ቁ Mଶ ሺ‫ݔ‬ሻ

And finally, since M/2(x) = 1 - M4(x), we have our result:

M(3x) = M(x)

ଷି଺Mర ሺ୶ሻିMఴ ሺ୶ሻ

ଵା଺Mర ሺ୶ሻିଷMఴ ሺ୶ሻ

.

Now, with this understanding, let us explore the construction on the Lemniscates. The point over the Lemniscates with respect to the arc length s could be constructed by straightedge and compass iffb=M(s) is a constructible number. By Noting that as the Lemniscates be defined by the equation ሺ‫ ݔ‬ଶ ൅ ‫ ݕ‬ଶ ሻଶ ൌ ʹܽଶ ሺ‫ ݔ‬ଶ െ ‫ ݕ‬ଶ ሻ and that b2 = x2+y2, we note that b4 = x2 ± y2. Then by solving in terms of b, we see that:

ͳ ͳ ‫ ݔ‬ൌ rඨ ሺ‫ ݎ‬ଶ ൅ ‫ ݎ‬ସ ሻǢ ‫ ݕ‬ൌ rඨ ሺ‫ ݎ‬ଶ െ ‫ ݎ‬ସ ሻ ʹ ʹ V. THE CRYPTOGRAPHIC SCHEME ON LEMNISCATES CURVE WITH REFERENCE TO ECC The discussion in previous section are by considering the numbers on the set of all real numbers. Usually real number operations comparatively slow with integer number operations. Also sometimes the result will not be accurate due to the round-off error in Real numbers. Operations in Cryptographic scheme should be accurate and of course faster. To achieve this, we need to consider the points of the curve over the prime field FP. We select the field in such a way that it should finitely large number of points suitable for cryptographic operations.

B. Domain parameter over Fp The Fpdomain parameters are a 5-tuple T=(p,a,G,n,h); where p± the prime number for the finite field Fp, a± the parameter of the curve, G± the generator point (xG,yG), a point on this curve which have been chosen for Cryptographic operations, n± the order of the curve. The scalar for point multiplication is chosen from a number between 0 and n ± 1, h± the cofactor where h = #F(Fp)/n.#F(Fp), is the total number of points taken on the Lemniscates curve. C. The Key Pairs In public key cryptosystem for secure transmission key exchange is used without considering any third parties. The key exchange scheme is described here.For a chosen Lemniscates domain parameters T=(p,a,G,n,h), or T=(m,f(x),a,G,n,h), a Lemniscates curve key pair (d,Q) related with Tcontainsa secrete key d, which is taken from the interval [1,n ± 1], and a public key Q=(xQ, yQ) which is actually the point Q=dG. D. Encryption and Decryption To send anencrypted message Pm to a receiver, say B, the sender, say A, select a random positive wholenumberk and results the cipher text Cmcontaining the pair of points. Cm=[kG, Pm+kPB] Here,Aused B¶VSXEOLF NH\ PB. To get back the original plain text from the cipher text, B find the product of the first point in the pair andB¶VSULYDWH key nBthen subtracts the result from the second point as given below: Pm+kPB±nB(kG)=Pm+k(nBG) ± nB(kG)=Pm Key sharing between the users A&B can be dond as follows: 1. A select nA, an integer