On RSA moduli with almost half of the bits prescribed

Abstract

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 = pl, where p and l are primes, with prescribed bit patterns. We are now able to specify about n bits instead of about n/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.

A Fourier-analytic approach to counting partial Hadamard matrices

“…For ε < 1/4, this term approaches zero. Indeed, for ε = 1/8 and t > n 8 , we have for n sufficiently large U(n, t, t Combining this with (58), for t > n 8 and n sufficiently large, we have U(n, t) := min Here we took a = δ 2 − (nδ) 4 and b = (nδ) 6 . So, if there is a solution, we must have…”

“…4t ) ≥ Re(ψ(γ)) 4t 1 + (nδ) 4 . Notice that this estimate is ideal when we need an estimate for powers of ψ(γ).…”

On RSA moduli with half of the bits prescribed

“…From m = n − n 3 4 ln n > 0, one deduces that n 2 13 . Thus the result of [1] makes sense for n 2 13 .…”

“…In [1], it is claimed that for m = n − n 3 4 ln n and any binary string σ of length m, the algorithm RSA-Modulus(n, m, σ ) terminates in expected polynomial time. From m = n − n 3 4 ln n > 0, one deduces that n 2 13 .…”

“…In [1] and [5], an algorithm named RSA-Modulus(n, m, σ ) was given to construct such kind of moduli M ∈ M n . However the number of bits prescribed in advance is less than the heuristic algorithms in [3] and [6].…”