RSA and a higher degree diophantine equation

Let N = pq be an RSA modulus, i.e the product of two large primes p and q. Without loss of generality, we assume that q < p. Morever, throughout this paper we assume that the primes p and q are balanced, in other words, that the bitsizes of the primes are equal so that q < p < 2q. Let e, d be the public and secret exponents satisfying ed ≡ 1 (mod φ(n)) where φ(n) = (p−1)(q−1) is the Euler totient function. To speed up the RSA decryption of some devices with limited computing power such as smart card, one might be tempted to use short secret exponents d. In 1990, Wiener [11] showed that if d < 13N 1 4 , then RSA was insecure. Wiener’s method is based on approximations using continued fractions. Verheul and van