On the Strength of the Strong RSA Assumption

The strong RSA assumption is an assumption that the following problem is hard to solve: Given an RSA modulus and a ciphertext, find a pair of plaintext and exponent corresponding to them. It differs from the standard RSA assumption in a sense that in the strong version, no exponent is given as an input. The strong RSA assumption is considered to be stronger than the RSA assumption, but their exact relationship is not known. We investigate the strength of the strong RSA assumption and show that the strong RSA assumption restricted to low exponents is equivalent to the assumption that RSA problem is intractable for any low exponent. We also show that in terms of algebraic computation, the strong RSA assumption is properly stronger than the RSA assumption if there exists an RSA modulus n such that gcd(φ(n), 3) = 1 and RSA problem is intractable.