| Summary: | This paper reports four new cryptanalytic attacks which show that the t instances of RSA moduli N = pq can be simultaneously factored in polynomial time using simultaneous diophantine approximations and lattice basis reduction techniques. In our technique we utilize the relation given by N−[(a j/i+b j/I / (2ab) j/2i + a 1/j+b 1/j / (2ab) 1/2j) √N] + 1 as a good approximations of Φ (N) for unknown positive integers d, di, ki, k, and zi. We construct four system of equations of the form esd − ksΦ(Ns) = 1, esds − kΦ (Ns) = 1, esd − kΦ (Ns) = zs and esds − kΦ (Ns) = zs where s = 1, 2, ..., t. In our attacks, we improve the short decryption exponent bounds of some reported attacks.
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