I'm using TLS with a Diffie-Hellman key exchange to negotiate the symmetric keys. How long should the exponents be?
Diffie-Hellman exponents need at a size at least twice the security level.
So for elliptic curve DH you need the same size as the underlying field. For example 256 bits for P-256.
For finite-field DH you still need twice the security level as exponent, but significantly larger fields. For example for an 80 bit level you need 160 bit exponents and a ~1024 bit field.