On RSA moduli with almost half of the bits prescribed

We show that using character sum estimates due to H. Iwaniec leads to an improvement of recent results about the distribution and finding RSA moduli M=plM=pl, where pp and ll are primes, with prescribed bit patterns. We are now able to specify about nn bits instead of about n/2n/2 bits as in the previous work. We also show that the same result of H. Iwaniec can be used to obtain an unconditional version of a combinatorial result of W. de Launey and D. Gordon that was originally derived under the Extended Riemann Hypothesis.