Recently new types of differential power analysis attacks (DPA) against elliptic curve cryptosystems (ECC) and RSA systems have been introduced. Most existing countermeasures against classical DPA attacks are vulnerable to these new DPA attacks which include refined power analysis attacks (RPA), zero-value point attacks (ZPA), and doubling attacks. The new attacks are different from classical DPA in that RPA uses a special point with a zero-value coordinate, while ZPA uses auxiliary registers to locate a zero value. So, Mamiya et al proposed a new countermeasure against RPA, ZPA, classical DPA and SPA attacks using a basic random initial point. His countermeasure works well when applied to ECC, but it has some disadvantages when applied to general exponentiation algorithms (such as RSA and ElGamal) due to an inverse computation. This paper presents an efficient and improved countermeasure against the above new DPA attacks by using a random blinding concept on the message different from Mamiya's countermeasure and show that our proposed countermeasure is secure against SPA based Yen's power analysis which can break Coron's simple SPA countermeasure as well as Mamiya's one. The computational cost of the proposed scheme is very low when compared to the previous methods which rely on Coron's simple SPA countermeasure. Moreover this scheme is a generalized countermeasure which can be applied to ECC as well as RSA system.