Paper 2002/013

Generic Lower Bounds for Root Extraction and Signature Schemes in General Groups

Ivan Damgard and Maciej Koprowski


We study the problem of root extraction in finite Abelian groups, where the group order is unknown. This is a natural generalization of the problem of decrypting RSA ciphertexts. We study the complexity of this problem for generic algorithms, that is, algorithms that work for any group and do not use any special properties of the group at hand. We prove an exponential lower bound on the generic complexity of root extraction, even if the algorithm can choose the "public exponent" itself. In other words, both the standard and the strong RSA assumption are provably true w.r.t. generic algorithms. The results hold for arbitrary groups, so security w.r.t. generic attacks follows for any cryptographic construction based on root extracting. As an example of this, we revisit Cramer-Shoup signature scheme. We modify the scheme such that it becomes a generic algorithm. This allows us to implement it in RSA groups without the original restriction that the modulus must be a product of safe primes. It can also be implemented in class groups. In all cases, security follows from a well defined complexity assumption (the strong root assumption), without relying on random oracles, and the assumption is shown to be true w.r.t. generic attacks.

Note: An error in the security proof of the signature scheme was corrected.

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. Extended version of what is to appear in Eurocrypt 2002.
root extraction problemRSAdigital signatures
Contact author(s)
mkoprow @ brics dk
2004-02-24: last of 2 revisions
2002-01-29: received
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      author = {Ivan Damgard and Maciej Koprowski},
      title = {Generic Lower Bounds for Root Extraction and Signature Schemes in General Groups},
      howpublished = {Cryptology ePrint Archive, Paper 2002/013},
      year = {2002},
      note = {\url{}},
      url = {}
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