Communications of the ACM
Algorithmic number theory
Finite fields
The Relationship Between Breaking the Diffie--Hellman Protocol and Computing Discrete Logarithms
SIAM Journal on Computing
Towards the Equivalence of Breaking the Diffie-Hellman Protocol and Computing Discrete Algorithms
CRYPTO '94 Proceedings of the 14th Annual International Cryptology Conference on Advances in Cryptology
A Provably Secure Additive and Multiplicative Privacy Homomorphism
ISC '02 Proceedings of the 5th International Conference on Information Security
Non-Interactive CryptoComputing For NC1
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
On The Complexity Of Matrix Group Problems I
SFCS '84 Proceedings of the 25th Annual Symposium onFoundations of Computer Science, 1984
Lower bounds for discrete logarithms and related problems
EUROCRYPT'97 Proceedings of the 16th annual international conference on Theory and application of cryptographic techniques
Abstract models of computation in cryptography
IMA'05 Proceedings of the 10th international conference on Cryptography and Coding
Factoring Polynomials over Special Finite Fields
Finite Fields and Their Applications
On Black-Box Ring Extraction and Integer Factorization
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part II
Fully homomorphic encryption using ideal lattices
Proceedings of the forty-first annual ACM symposium on Theory of computing
On the Analysis of Cryptographic Assumptions in the Generic Ring Model
ASIACRYPT '09 Proceedings of the 15th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
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The black-box field (BBF) extraction problem is, for a given field F, to determine a secret field element hidden in a black-box which allows to add and multiply values in F in the box and which reports only equalities of elements in the box. This problem is of cryptographic interest for two reasons. First, for F = Fp it corresponds to the generic reduction of the discrete logarithm problem to the computational Diffie-Hellman problem in a group of prime order p. Second, an efficient solution to the BBF extraction problem proves the inexistence of field-homomorphic one-way permutations whose realization is an interesting open problem in algebra-based cryptography. BBFs are also of independent interest in computational algebra. In the previous literature BBFs had only been considered for the prime field case. In this paper we consider a generalization of the extraction problem to BBFs that are extension fields. More precisely we discuss the representation problem defined as follows: For given generators g1,..., gd algebraically generating a BBF and an additional element x, all hidden in a black-box, express x algebraically in terms of g1,..., gd. We give an efficient algorithm for this representation problem and related problems for fields with small characteristic (e.g. F = F2n for some n). We also consider extension fields of large characteristic and show how to reduce the representation problem to the extraction problem for the underlying prime field. These results imply the inexistence of field-homomorphic (as opposed to only group-homomorphic, like RSA) one-way permutations for fields of small characteristic.