Linear congruential random number generators must have large moduli to attain maximum periods, but this creates integer overflow during calculations. Several methods have been suggested to remedy this problem while obtaining portability. Approximate factoring is the most common method in portable implementations, but there is no systematic technique for finding appropriate multipliers and an exhaustive search is prohibitively expensive. We offer a very efficient method for finding all portable multipliers of any given modulus value. Letting M = AB+C, the multiplier A gives a portable result if B-C is positive. If it is negative, the portable multiplier can be defined as A = left perpendicularM/Brightperpendicular. We also suggest a method for discovering the most fertile search region for spectral top-quality multipliers in a two-dimensional space. The method is extremely promising for best generator searches in very large moduli: 64-bit sizes and above. As an application to an important and challenging problem, we examined the prime modulus 2(63)-25, suitable for 64-bit register size, and determined 12 high quality portable generators successfully passing stringent spectral and empirical tests.