Probability that the k-gcd of products of positive integers is B-smooth

In 1849, Dirichlet [5] proved that the probability that two positive integers are relatively prime is 1/ζ(2). Later, it was generalized into the case that positive integers has no nontrivial kth power common divisor. In this paper, we further generalize this result: the probability that the gcd of m products of n positive integers is B-smooth is ∏ p>B [ 1 − { 1 − ( 1 − 1 p )n}m] for m ≥ 2. We show that it is lower bounded by 1 ζ(s) for some s > 1 if B > n m m−1 , which completes the heuristic proof in the cryptanalysis of cryptographic multilinear maps by Cheon et al. [2]. We extend this result to the case of k-gcd: the probability is ∏ p>B [ 1 − { 1 − ( 1 − 1 p )n ( 1 + nH1 p + · · · + nHk−1 pk−1 )}m] , where nHi = (n+i−1 i ) .