Regarding the integer factorization problem, only one case is of particular interest for cryptography: the factorization of large semiprimes when the prime factors are nearly equal.
In this case, the most efficient method is to find the smallest integer n such that n^2 exceeds the semiprime. Any semiprime p*q will be less than some n^2 with p < n < q, and q << 2*n.
Suppose we want to factor 988027 and know it is semiprime. Its square root rounded to the nearest integer is 999. From trial multiplication, we find that 992 is the smallest integer that when squared is greater than 988027. And if we systematically check the numbers greater than 992 which end in 1, 3, 7, or 9, we quickly find that 997 is a factor of 988027.
The semiprimes used for cryptography are vastly larger, but the principle is the same. It makes more sense to use trial multiplication before trial division.
Trial multiplication is a method for solving the discrete logarithm problem. I see it as part of a one-two punch for the integer factorization problem.
