Abstract. Two closely-related pseudo-random sequence generators are presented: The 1/P-generator t with input P a prime, outputs the quotient digits obtained on dividing 1 by P. The x^2 mod N-generator with inputs N, x0 (where N = P-Q is a product of distinct primes, each congruent to 3 mod 4, and x0 is a quadratic residue mod N), outputs b0 b1 b2 . . . where b_i = parity(x_i) and x_{i+i} = x_i^2 mod N. From short seeds each generator efficiently produces long well-distributed sequences. Moreover, both generators have computationally hard problems at their core. The first generator's sequences, however, are completely inferable (from any small segment of 2|P|+1 consecutive digits one can infer the "seed," P), while the second, on the other hand, is cryptographically secure (no polynomial (| N | )-time statistical test can distinguish such sequences from ran dom uniformly-distributed sequences). The second generator has additional interesting properties: from knowledge of x0 and N but not P or Q, one can generate the sequence forwards but not backwards. From the additional knowledge of P and Q, one can generate the sequence backwards. Yet more knowledge about N, including the factors of P—1 and Q - 1, enable one to "jump" about from any point in the sequence to any other. Because of these properties, the x2 mod N-generator promises many interesting applications, e.g., to public- key cryptography. To use these generators in practice, an analysis is needed of various properties of these sequences such as their periods. This analysis is begun here.