Correcting Errors in Private Keys Obtained from Cold Boot Attacks

Abstract

Based on the cold boot attack technique, this paper proposes a new algorithm to obtain the private key of the discrete logarithm (DL) based cryptosystems and the standard RSA from its erroneous value. The proposed algorithm achieves almost the square root complexity of search space size. More precisely, the private key of the DL based system with 160-bit key size can be recovered in 2^43.24 exponentiations while the complexity of the exhaustive search is 2^71.95 exponentiations if the error rate is given by 10%.

In case of the standard RSA with 1024-bit key size, our algorithm can recover the private key with 2^49.08 exponentiations if the error rate is given by 1%. Compared with the efficiency of some algorithms [7,6] to recover the private key in RSA using Chinese Remainder Theorem, the recoverable error rate of our algorithm is quite small. However, our algorithm requires only partial information of the private key d while other algorithms require additional information such as partial information of factors of the RSA modulus N.

The proposed algorithm can also be used for breaking countermeasure of differential power analysis attack. In the standard RSA, one uses the randomized exponent $\tilde{d}=d+r\cdot\phi(N)$ instead of the decryption exponent d with the random value r. When the size of a random value r is 26-bit, it can be shown that the randomized exponent can be recovered with 2^49.30 exponentiations if the error rate is 1%. Finally, we also consider the breaking countermeasure that splits the decryption exponent d into d ₁ and d ₂ of same size.