Power analysis can be used to attack many implementations of cryptosystems, e.g., RSA and ECC, and the doubling attack is a collision based power analysis performed on two chosen ciphertexts. In this paper, we introduced a modified doubling attack to threaten RSA and ECC implementations by exploiting only one chosen ciphertext of small order. To attack the RSA implementations we selected an input of order two while to attack the ECC implementations we exploited one chosen invalid point of small order on a cryptographically weak curve rather than on the original curve. We showed that several existing power analysis countermeasures for RSA and ECC implementations are still vulnerable to the proposed attack. To prevent the proposed attack, we suggested countermeasures for RSA as well as for ECC.