Black-Box Extension Fields and the Inexistence of Field-Homomorphic One-Way Permutations

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 problem proves the inexistence of certain field-homomorphic encryption schemes whose realization is an interesting open problems 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...

• Type: Conference
• Year: 2007
• Where: ASIACRYPT
• Authors: Ueli M. Maurer, Dominik Raub
• Keywords: ASIACRYPT 2007, Cryptology, Discrete Logarithm Problem, Extraction Problem, Representation Problem