In Eurocrypt’98 [1], Okamoto et al. exhibited a new trapdoor function based on the use of a special moduli (p²q) allowing easy discrete logarithm computations. The authors proved that the scheme’s resistance to chosen-plaintext attacks is equivalent to factoring n. Unfortunately, the proposed scheme suffers from not being a permutation (the expansion rate is ~ 3), and hence cannot be used for public-key signatures. In this paper, we show how to refine the function into a trapdoor permutation that can be used for signatures. Interestingly, our variant still remains equivalent to factoring and seems to be the second known trap-door permutation (Rabin-Williams’ scheme [3] being the first) provably as secure as a primitive problem.