Paper 2008/156

On Black-Box Ring Extraction and Integer Factorization

Kristina Altmann, Tibor Jager, and Andy Rupp


The black-box extraction problem over rings has (at least) two important interpretations in cryptography: An efficient algorithm for this problem implies (i) the equivalence of computing discrete logarithms and solving the Diffie-Hellman problem and (ii) the in-existence of secure ring-homomorphic encryption schemes. In the special case of a finite field, Boneh/Lipton and Maurer/Raub showed that there exist algorithms solving the black-box extraction problem in subexponential time. It is unknown whether there exist more efficient algorithms. In this work we consider the black-box extraction problem over finite rings of characteristic $n$, where $n$ has at least two different prime factors. We provide a polynomial-time reduction from factoring $n$ to the black-box extraction problem for a large class of finite commutative unitary rings. Under the factoring assumption, this implies the in-existence of certain efficient generic reductions from computing discrete logarithms to the Diffie-Hellman problem on the one side, and might be an indicator that secure ring-homomorphic encryption schemes exist on the other side.

Note: This is an extended version of the paper with the same title that appeared in the proceedings of ICALP 2008.

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. Unknown where it was published
Black Box Extraction ProblemInteger FactorizationHomomorphic Encryption
Contact author(s)
tibor jager @ rub de
2008-07-06: last of 2 revisions
2008-04-08: received
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      author = {Kristina Altmann and Tibor Jager and Andy Rupp},
      title = {On Black-Box Ring Extraction and Integer Factorization},
      howpublished = {Cryptology ePrint Archive, Paper 2008/156},
      year = {2008},
      note = {\url{}},
      url = {}
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