Magma online calculator gave the following [I was interested in CPU time after @fkraiem's answer].
TL;DR: a randomly chosen 200 bit probable prime exceeded CPU time limit of 120 seconds.
"is p prime?",IsPrime(p);
"Check: b^k = ",F!(b^k)," =?=", x;
is p prime? true
Check: b^k = 6361196924231058595008858273263807320 =?=
Calculations are restricted to 120 seconds.
Input is limited to 50000 bytes.
Running Magma V2.23-9.
Seed: 2717357355; Total time: 3.680 seconds; Total memory usage: 32.09MB.
Magma documentation on DL says:
The different kinds of finite fields in Magma are handled as follows (in this order):
Small Fields (any characteristic):
If the largest prime l dividing q - 1 is reasonably small (typically, less than 2^36), the Pohlig-Hellman algorithm is used (the characteristic p is irrelevant).
(b) Large Prime :
Suppose K is a prime field (so q=p). Then the Gaussian integer sieve [COS86], [LO91a] is used if p has at least 4 bits but no more than 400 bits, p - 1 is not a square, and one of the following is a quadratic residue modulo p: -1, -2, -3, -7, or -11.
If the Gaussian integer sieve cannot be used and if p is no more than 300-bits, then the linear sieve [COS86], [LO91a] is used. The precomputation stage always takes place and typically requires a lot more time than for computing individual logarithms (and may also require a lot of memory for large fields). Thus, the first call to the function Log below may take much more time than for subsequent calls.
Also, for large prime fields, in comparison to the Gaussian method the linear sieve requires much more time and memory than the Gaussian method for the precomputation stage, and therefore it is only used when the Gaussian integer algorithm cannot be used. See the example H27E3 in the chapter on sparse matrices for an explanation of the basic linear sieve algorithm and for more information on the sparse linear algebra techniques employed.
Taking this as a challenge let's try a 200 bit prime (probable prime)
First precomputation to find a generator:
p = 495998296780695342795805838124611282143307443742260790342431
a probable prime p found in 205 trials
Group order - Orders of these elements are
247999148390347671397902919062305641071653721871130395171215, 0 ]
a generator found in 4 trials
Take b=3, i.e., L;
this trial failed the CPU time limit
Note: I don't have access to my laptop which is at work, to try on the local copy of Magma.