Parallel supercomputer-based Monte Carlo and stochastic simulations require pseudorandom number generators that can produce distinct pseudorandom streams across many independent processes. We propose a scalable class of parallel and vectorizable pseudorandom number generators based on a non-cryptographic version of the RSA public-key exponentiation cipher. Our method generates uniformly distributed IEEE double-precision floating point pseudorandom sequences on $[0,1)$ by encrypting pseudorandom sequences of 64-bit integer messages by modular exponentiation. The advantages of the method are: the method is parallelizable by parameterization with each pseudorandom number generator instance derived from an independent 64-bit composite modulus, the method is fully scalable on massively parallel computing clusters because of the millions of available 32-bit prime numbers, the seeding and initialization of the independent streams is simple, the periods of the independent instances are all different and greater than $8.5\times 10^{37}$, and the method passes a battery of intrastream and interstream correlation tests. The calculations in each instance can be vectorized using steam splitting and can produce more than $10^8$ pseudorandom numbers per second on each multicore CPU.