In 2022, Cotan and Teşeleanu proposed a new RSA-like cryptosystem, where the modulus is a classical RSA integer of the form \(N=pq\) , while the public exponent e and the private exponent d satisfy \(ed\equiv 1\pmod {\psi _n(N)}\) with \(\psi _n(N)=\frac{(p^n-1)(q^n-1)}{(p-1)(q-1)}.\) At Africacrypt 2024, Nitaj, Adenan, and Ariffin presented an attack on this variant when \(d\equiv \frac{1}{e}\pmod {\psi _n(N)}\) is small. In this paper, we study the more general situation where the public exponent is of the form \(e\equiv \frac{z}{u} \pmod {\psi _n(N)}\) . We transform this equation into one of the form \(xH(y)+z\equiv 0\pmod e\) , and, using Coppersmith’s technique and lattice basis reduction, we present a method to solve it when the variables x, y and z are suitably small. As a byproduct, we show that the scheme of Cotan and Teşeleanu is vulnerable for more classes of public exponents, and that our attack extends several former attacks on this RSA variant.
