Near-uniform sampling of factored smooth integers in expected polynomial time

Given integers \(2≤ y≤ X\), we consider the generation of an integer chosen nearly uniformly from the \(y\)-smooth integers at most \(X\), together with its complete prime factorization. We give an unconditional algorithm using fair random bits whose output law is within total variation \(2⁻ʰ\) of uniform and whose expected bit complexity is bounded by a fixed polynomial in \(log_2 X+h\), uniformly in \(y\). The algorithm requires no prime list, smooth-number counting oracle, saddle-point evaluation or general integer-factorization routine. Its construction combines tilted independent geometric prime exponents with sparse proposals over dyadic integer intervals. An explicitly filtered grid contains a useful tilt without computing it. Uniform smooth-number estimates give polynomial acceptance, and certified finite-precision primitives preserve both the distributional guarantee and the expected work bound. Every returned integer and its displayed prime factorization are exact; the approximation concerns only the sampling distribution. The guarantee concerns expected bit complexity; no practical running-time comparison is asserted.