More constructing pairing-friendly elliptic curves for cryptography

(1) q(x) = p(x) d for some d 1 and p(x) that represents primes. (2) r(x) = c ~ r(x) with c 2 Z 1 and ~ r(x) that represents primes. (3) r(x) j q(x) + 1 t(x). (4) r(x) j k (t(x) 1), where k is the kth cyclotomic polynomial. (5) 4q(x) t(x) 2 = Dy 2 has infinitely many integer solutions (x; y). Assume p D 2 Q( k ). If k (u(x)) is reducible with a factor of degree '(k) for some u(x) 2 Q(x), we can take r(x) to be one of its irreducible factor. To obtain such u(x), it is necessary and sufficient that u(a(x)) x (mod k (x)) for some a(x) 2 Q(x). we consider the case 1 a 0 a 0 2 a 2 2 2a 1 a 3 a 0 3 3a 2 (a 0 a 2 + a 1 2 a 3 2 ) 6a 0 a 1 a 3 0 a 1 2a 0 a 1 2a 2 a 3 a 3 3 3a 1 (a 1 a 3 + a 2 2 a 0 2 ) 6a 0 a 2 a 3 0 a 2 a 1 2 a 3 2 + 2a 0 a 2 a 2 3 + 3a 0 (a 0 a 2 + a 1 2 a 3 2 ) 6a 1 a 2 a 3 0 a 3 2a 1 a 2 + 2a 0 a 3 a 1 3 3a 3 (a 1 a 3 + a 2 2 a 0 2 ) + 6a 0 a 1 a 2 1 C C C A : Let d and n i be as follows: d := (a 1 2 + a 3 2 )((a 1 a 3 ) 2 + 2a 2 2 )((a 1 + a 3 ) 2 2a 2 2 ); n 0 := a 2 (5a 1 4 a 3 5a 1 3 a 2 2 + 5a 1 a 2 2 a 3 2 2a 2 4 a 3 + 3a 3 5 ); n 1 := a 1 5 4a 1 3 a 3 2 + 9a 1 2 a 2 2 a 3 + a 1 (2a 2 4 + 3a 3 4 ) + 3a 2 2 a 3 3 ; n 2 := a 1 3 a 2 + 3a 1 a 2 a 3 2 2a 2 3 a 3 ; n 3 := a 3 3 a 1 2 a 3 + 2a 1 a 2 2 :