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Correcting Errors in RSA Private Keys

Wilko Henecka, Alexander May, Alexander Meurer

Abstract:
Let pk=(N,e) be an RSA public key with corresponding secret key sk=(p,q,d,d_p,d_q, q_p^{-1}). Assume that we obtain partial error-free information of sk, e.g., assume that we obtain half of the most significant bits of p. Then there are well-known algorithms to recover the full secret key. As opposed to these algorithms that allow for correcting erasures of the key sk, we present for the first time a heuristic probabilistic algorithm that is capable of correcting errors in sk provided that e is small. That is, on input of a full but error-prone secret key sk we reconstruct the original sk by correcting the faults.

More precisely, consider an error rate of delta in [0,\frac 1 2), where we flip each bit in sk with probability delta resulting in an erroneous key sk'. Our Las-Vegas type algorithm allows to recover sk from sk' in expected time polynomial in log N with success probability close to 1, provided that delta < 0.237. We also obtain a polynomial time Las-Vegas factorization algorithm for recovering the factorization $(p,q)$ from an erroneous version with error rate delta < 0.084.

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