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**2005/024**

**Title:**

Weak keys of the Diffe Hellman key exchange I

**Authors:**
A. A. Kalele, V. R. Sule

**Abstract:**

This paper investigates the Diffie-Hellman key exchange scheme

over the group \fpm^* of nonzero elements of finite fields and

shows that there exist exponents k, l satisfying certain

conditions called the \emph{modulus conditions}, for which the

Diffie Hellman Problem (DHP) can be solved in polynomial number

of operations in m without solving the discrete logarithm problem (DLP). These special private keys of the scheme are termed

\emph{weak} and depend also on the generator a of the cyclic group. More generally the triples (a,k,l) with generator

a and one of private keys k,l weak, are called \emph{weak triples}. A sample

of weak keys is computed and it is observed that their number may not be

insignificant to be ignored in general. Next, an extension of the

analysis and weak triples is carried out for the Diffie Hellman

scheme over the matrix group \gln and it is shown that for an

analogous class of session triples, the DHP can be solved without

solving the DLP in polynomial number of operations in the matrix

size n. A revised Diffie Hellman assumption is stated, taking into account the above exceptions.

**ePrint:**
https://eprint.iacr.org/2005/024

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