Diffie Hellman Key Exchange (Finding # of bits/digits of secret key)
Assuming that $g^A \bmod p$  and $g^B \bmod p$ are 100 digits long, would Charlie be able to determine a bound (in terms of digits or bits) for the respective secret key $A$ and $B$ ?


100 digits is approximately 332 bits; if $p$ is 332 bits long, then it is known to be feasible to apply the Number Field Sieve algorithm to recover the values $A$ and/or $B$; that certainly would give him the bound.

For reference, Antoine Joux and Reynald Lercier computed the discrete log of a 431 bit (130 digit) base in three weeks.