Question

CYRIPTOLOGY

An elliptic curve defined on Z29 E: y2 = xº + 4x + 20 mod (29) It is defined as. If P = (1.5) is a point on this curve, which is 2P?

Answer 1

An elliptic Curve defined on Zz9 E; Y? –X? +ux +20 mod (29 a point on Cune. P = (1,5) a y² » Elliptic is of form Exp+ ax+b mode of a elliptic Curve in xp , then eq, is as above. Here x, y, a and b are all within tp./i.e. they are integers ) mod p) Curre fulfill this condition ya +276?+0. Coebb all ar on So, our Eq, is y²= x² + 4x+20 mod 29 Hene ar observed p=29 (au 229 Now, y² => y? -x3+4x +20 Given point p (1,5) is on Curve. a first here a=4 and b=20 Check for Condition 4a3+276²20 4(43) +27(20*) 70 Now given plis) is on curve 5 3 = 1+ 4 (1) +20 5² = 1+ 4 +20 = 25=25 LHS - RHS So, a point is present on Curve. CSScanned with CamScanner

on curve and we Now, we have to find 2p. p is a point have to find third intersection point , that neglect a crou -axis so that it result 2p poisk. → first find the tangent dine eq. for the curve dy - slope of line da So, y=x0+4x+ 20 d(72) 3x²+4 da =) 24.dy 3x²74 de dy = x²74 de 24 So, dy at (1,5) is 3+4 11 7 dx 10 10 So, our tangent dine is (8,5) (y-yı)m (2-0) (y-5)= =( 7(x-1) loy-50 – 74-7 74-10+43 = = 0 from this: 7x+43 =104 y = 72+43 10 Suboy in Eurve Eq CSScanned with CamScanner

Gx+43) x ² 4x +20 10 (72+43)*= 100/x*tu2+20) 49x?+1849+602x = 100x37 400X + 2000 => 100x3 -49x?+400% -602X +2000-1849.00 get So, here after Solving 100x3– 49x2 - 2026151.50 this equation we X,= -1.5 , X2 = 1+i*, X3 = 1-;*0 So, the third intersection point is at x = -)5. Calculating y=7&+43 10 y=7(-1.5)+43 10 > y = - 10:5+43 10 = 32.5/10 = 3:25 So, the thind fén tersection point is at (-1.5, 3.25) i-e. (1,5) + (1,5) = (-1.5,3. 3.25). P+p=2p CSScanned with CamScanner