The square root Diffie-Hellman problem

Many cryptographic schemes are based on computationally hard problems. The computational Diffie-Hellman problem is the most well-known hard problem and there are many variants of it. Two of them are the square Diffie-Hellman problem and the square root Diffie-Hellman problem. There have been no known reductions from one problem to the other in either direction. In this paper we show that these two problems are polynomial time equivalent under a certain condition. However, this condition is weak, and almost all of the parameters of cryptographic schemes satisfy this condition. Therefore, our reductions are valid for almost all cryptographic schemes.