On the Improvement of the BDF Attack on LSBS-RSA

Abstract

An $\left( \alpha ,\beta ,\gamma \right)$-LSBS RSA denotes an RSA system with primes sharing α least significant bits, private exponent d with β least significant bits leaked, and public exponent e with bit-length γ. Steinfeld and Zheng showed that LSBS-RSA with small e is inherently resistant to the BDF attack, but LSBS-RSA with large e is more vulnerable than standard RSA. In this paper, we improve the BDF attack on LSBS-RSA by reducing the cost of exhaustive search for k, where k is the parameter in RSA equation: $ed=k\cdot \varphi \left( N\right) +1$. Consequently, the complexity of the BDF attacks on LSBS-RSA can be further reduced. Denote σ as the multiplicity of 2 in k. Our method gives the improvements, which depend on the two cases:

• 1 In the case $\gamma \leq \min \left\{ \beta ,2\alpha \right\} -\sigma$, the cost of exhaustive search for k in LSBS-RSA can be simplified to searching k in polynomial time. Thus, the complexity of the BDF attack is independent of γ, but it still increases as α increases.

• 1 In the case $\gamma >\min \left\{ \beta ,2\alpha \right\} -\sigma$, the complexity of the BDF attack on LSBS-RSA can be further reduced with increasing α or β.

More precisely, we show that an LSBS-RSA is more vulnerable under the BDF attack as $\max \left\{ 2\alpha ,\beta \right\}$ increases proportionally with the size of N. In the last, we point out that although LSBS-RSA benefits the computational efficiency in some applications, one should be more careful in using LSBS-RSA.