void XTR_FindPrimesAndGenerator(RandomNumberGenerator &rng, Integer &p, Integer &q, GFP2Element &g, unsigned int pbits, unsigned int qbits) { assert(qbits > 9); // no primes exist for pbits = 10, qbits = 9 assert(pbits > qbits); const Integer minQ = Integer::Power2(qbits - 1); const Integer maxQ = Integer::Power2(qbits) - 1; const Integer minP = Integer::Power2(pbits - 1); const Integer maxP = Integer::Power2(pbits) - 1; Integer r1, r2; do { bool qFound = q.Randomize(rng, minQ, maxQ, Integer::PRIME, 7, 12); assert(qFound); bool solutionsExist = SolveModularQuadraticEquation(r1, r2, 1, -1, 1, q); assert(solutionsExist); } while (!p.Randomize(rng, minP, maxP, Integer::PRIME, CRT(rng.GetBit()?r1:r2, q, 2, 3), 3*q)); assert(((p.Squared() - p + 1) % q).IsZero()); GFP2_ONB<ModularArithmetic> gfp2(p); GFP2Element three = gfp2.ConvertIn(3), t; while (true) { g.c1.Randomize(rng, Integer::Zero(), p-1); g.c2.Randomize(rng, Integer::Zero(), p-1); t = XTR_Exponentiate(g, p+1, p); if (t.c1 == t.c2) continue; g = XTR_Exponentiate(g, (p.Squared()-p+1)/q, p); if (g != three) break; } assert(XTR_Exponentiate(g, q, p) == three); }