Attacking an RSA-like Cryptosystem Using Continued Fractions and Lattices

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Let $N = pq$ be the product of two balanced primes. Cotan and Te\c seleanu (2023) introduced a family of RSA-like cryptosystems defined by $ed - k(p^n - 1)(q^n - 1) = 1$, where $n \geq 1$, encompassing classical RSA ($n=1$) and the Elkamchouchi–Elshenawy–Shaban variant ($n=2$). We present a new attack for $n=3$ that integrates continued fractions with lattice-based methods, naturally extending previous results for $n = 1, 2, 4, 6$.