These cryptanalytic results (see cryptanalysis) show that the RSA public key encryption scheme is not secure under certain conditions on the RSA parameters.
To use RSA one first chooses two large prime numbers p and q. The public modulus is N = pq, and p and q are kept secret (see modular arithmetic). A public exponent e and a secret (private) exponent d are chosen satisfying the RSA equation ed = 1 mod(N − p − q + 1). Typically, the secret exponent d is of approximately the same bit-length as the public modulus N, the public exponent e has a bit-length no greater than the bit-length of N, and the primes p and q are approximately the same bit-length. Because RSA decryption and signature generation depend on the length of the secret exponent d, it is tempting to use a short d to speed up these operations. However, in 1990, Wiener [1 showed that if the length of d is less than one-quarter of the length of N, then there is an efficient attack which computes the secret exponent dfrom the...
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References
Wiener, M. (1990). “Cryptanalysis of short RSA secret exponents.” IEEE Transactions on Information Theory, 36 (3), 553–558.
Coppersmith, D. (2001). “Finding small solutions to small degee polynomials.” Proceedings of Cryptography and Lattices Conference, Lecture Notes in Computer Science, vol. 2146, ed. J.H. Silverman. Springer-Verlag, Berlin.
Boneh, D. and G. Durfee (2000). “Cryptanalysis of RSA with private key d less than N 0.292.” IEEE Transactions on Information Theory, 46 (4), 1339–1349.
Durfee, G. and P. Nguyen (2000). “Cryptanalysis of RSA schemes with short secret exponent from Asiacrypt'99.” Proceedings of ASIACRYPT 2000, Lecture Notes in Computer Science, vol. 1976, ed. T. Okamoto. Springer-Verlag, Berlin, 14–29.
May, A. (2002). “Cryptanalysis of unbalanced RSA with small CRT-exponent.” Proceedings of CRYPTO 2002, Lecture Notes in Computer Science, vol. 2442, ed. M. Yung. Springer-Verlag, Berlin, 242–256.
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Durfee, G. (2005). Wiener, Boneh–Durfee, and May Attacks on the RSA Public Key Cryptosystem. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_460
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