Question

1. List all points (x,y) in the elliptic curve y2≡ x3 + 2x - 9 (mod 19). (Hint: Corresponding to any given x , points (x,y) and (x,-y) can exist on the elliptic curve only if y2≡ x3 + 2x - 9 (mod 19) is a quadratic residue mod 19. Recall that a value v ∊ Zp is a quadratic residue modulo p only if v(p-1)/2≡ 1 (mod p). If v is indeed a quadratic residue, we can calculate the two square roots of v as ±v(p+1)/4(mod p), if p≡ 3 (mod 4) -- which is the case for p=19 as 19 = 3+4.4 )

3. List all points (x,y) in the elliptic curve y2E x3 + 2x - 9 (mod 19). (Hint: Corresponding to any given x , points (x,y) and (x,-y) can exist on the elliptic curve only if yx3+2x-9 (mod 19) is a quadratic residue mod 19. Recall that a value ve Zp is a quadratic residue modulo p only if vip- 1 (mod p). If v is indeed a quadratic residue, we can calculate the two square roots of v as tvip(mod p), if p 3 (mod 4)- which is the case for p-19 as 19-3+4.4)

Answer 1

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