We present several attacks on RSA that factor the modulus in polynomial time under the condition that a fraction of the most significant bits or least significant bits of the private exponent is available to the attacker. Our new attacks on RSA are the first attacks of this type that work up to full size public or private exponent.

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Wiener's famous attack on RSA with d < N 0.25 shows that using a small d for an efficient decryption process makes RSA completely insecure. As an alternative, Wiener proposed to use the Chinese Remainder Theorem in the decryption phase, where dp = d mod (p − 1) and dq = d mod (q − 1) are chosen significantly smaller than p and q. The parameters dp, dq are called private CRT-exponents. Since Wiener's proposal in 1990, it has been a challenging open question whether there exists a polynomial time attack on small private CRT-exponents. In this paper, we give an affirmative answer to this question, and show that a polynomial time attack exists if dp and dq are smaller than N 0.073 .

Abstract. We describe an attack on the RSA cryptosystem when the private exponent d is chosen to be 'small', under the condition that a sufficient amount of bits of d is available to the attacker. The attack uses a 2-dimensional lattice and is therefore (in the area of the keyspace where it applies) more efficient than known attacks using Coppersmith techniques. Moreover, we show that the attacks of Wiener and Verheul/Van Tilborg, using continued fractions techniques, are special deterministic cases of our attack, which in general is heuristic.

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