Question

Q1. Consider the curve y2 = x3 + 3x + 5 mod 17. (i) (ii) (iii) (iv) (v) (vi) Confirm that it is an elliptic curve. Determine three points on the curve over real numbers Determine three points on the curve over integer numbers For one of the points P in (iii), find 2P (or double) For two of the points P and Q in (iii), find P+Q Find the bound for the number of points on this curve using Hesse's theorem.

Answer 1

Solution :> given curve is y= 33+32 +5 mod 17 - Noce, general equation of ellipti curve is 43= 23+ ax + b. -> ☺ so equation 0 is look like as equation 1 . Now it is enough to check that . equation @ has does not have any Singularities. This can be done by computing the discriminant. A = -16 (493+2763). Rule :-) d curve of equation @ is elliptic I it it is non singular and it is non-sigular it and only if sto.

I 1. as comparing 0 and a = 3_ 65 - DE- 16(4x(3)37 27x65) 2) ---------- = -16 (108 +675) = -16 X 783 - 12528 is '-12595 = 0 (mod 17) is not possible : 0 is given closurve is an elliptical Clerve NOCE_We determine_Houde_ points on the curve over real line. so given equation is __-42222 + 3x +5 mod 17 = y² (x73x+5) is divisible by 17 ->__ Y2_ 22-32 -5 = 17k for some kez set k=1 3 4282-3x - 5 = 17. => 92 = 22+ 32 +5+17 = y² = x² + 3x+22. 11 X=0 - y = 0+30 +22 = 22 Y = V22 = 02+30) + 29 = 1 +3 +22 = y² = 26 > y = 126

Ret = 2 -> >> => ? 7 = (2) +868) + 22. y = 4 + 6 +22 = 10+92 g = √32 i are Ibrece points on cloves wall over seal numbers (0,122), (1,826), (2, 32). - Three points on the cleave over integer framblys considered __ 42= 3 + 3x+22 take square to both side. I try so many method to find points on the curve over integer but there is no point such that both n and y are integer satisfying above .600 equation we are fin ap for (1) but there is no points in W. Do we find 2p_408 @ as followed het p = (0,822) or p=(1,826). => 2P = 210, 22) -> 28= (2,2126) = (0,2172: ③ for find P+Q . P = (0, 122) and Q = (1, 826) __ = PPQ = (0,152) + (1, V26) = (1, Vz2+V2

Date: Pege No. Accogiding to Hesses throuen for bound for Number of this elliptical curve is formula points on _IN - (2+1) 14 2 VI where N is the curve cold over & elements .. s her mod 17 number of points on finite field e with the means value of g = 17 => bound formula is IN-117+12] 5 2117 IN -18/5 2VT. ☺ -- equation ©. guepaesent Hoe formula for N number the curve . = X oequired of points bound on