Given a square-free integer ∆ < 0, we present an algorithm constructing a pair of primes p and q such that q|p + 1 − t and 4p − t 2 = ∆f 2 , where |t| ≤ 2 √ p for some integers f, t. Together with a CM method presented in the paper, such primes p and q are used for a construction of an elliptic curve E over a finite field F p such that the order of E is divisible by a large prime. It is shown that our algorithm works in polynomial time.
We present a general method of generating primes p and q such that q divides Φ n (p), where n > 2 is a fixed number. In particular, we present the deterministic method of finding a primitive nth roots of unity modulo q. We estimate the computational complexity of our methods.Keywords: primes of a special form, computing a primitive nth root of unity modulo primes, public key cryptography.
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