The security of the Diffie-Hellman key exchange lies in the fact that, while it is relatively easy to calculate exponentials modulo a prime, it is very easy to calculate discrete logarithms.

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The security of the Diffie-Hellman key exchange actually relies on the difficulty of calculating discrete logarithms modulo a prime, not the ease of it. Calculating exponentials modulo a prime is indeed relatively easy, which is necessary for the protocol to work efficiently, but the hard problem that secures it is the discrete logarithm problem.

The Security of the Diffie-Hellman Key Exchange Lies in the Fact