If cryptanalysts today were to crack the original Enigma machine, “how fast” or “how easily” could they do it? What methods would they use?
The original cracking was significantly helped by operator mistakes, like always including the same header, always including a weather report, and finally, a long message of LLLLLLLL… being sent. Let's assume no such operator mistakes happened.
What methods would they use?
Since WW2, we know the security of Enigma machines was weakened by the reflector, resulting in two problems:
Also, history has shown that Enigma had another security problem: the contradiction to Kerckhoffs' principle "The security of a system should not depend on the privacy of the algorithm. It should only be based on the secrecy of the key.".
Enigma got it wrong because:
(source: https://www.infsec.cs.uni-saarland.de/teaching/SS07/Proseminar/slides/maurer-Enigma.pdf)
Since you're asking about today, I will therefore assume the fact that we know about the internal workings of Enigma since WW2.
Based on your assumptions and the known Enigma weaknesses, one of the well-known attacks to break Enigma is using the Index of Coincidence (short: IC), a statistical measure of text which distinguishes text encrypted with a substitution cipher from plain text.
Let's put Enigma into the formula: $IC = \frac{{\textstyle \sum_{i=0}^{25} } f_i(f_i-1)}{n(n-1)}$ where $x_1,...,x_n$ is the string of letters and $f_0,...,f_{25}$ the frequency of the letters in that string. For a random string that boils down to $IC ≈ 0.038$ and for a natural language $IC ≈ 0.065$.
I'll skip the details and simply give you some raw numbers while summing up how IC can be used for an attack on Enigma:
how fast/ how easily could they do it?
Generally, that's hard to say without having any specific information about available resources you are assuming an attacker would have.
A single computer might take a little while depending on how old it is (in terms of CPU power), but things like distributed networks will probably make you cry by successfully attacking Enigma in a few blinks of an eye. If you happen to have any specific computing setups in mind, you can take the provided number of operations mentioned above and simply do the related maths to calculate the related speeds.
And last but not least - to answer the title of your question too...
How cryptographically secure was the original WW2 Enigma machine, from a modern viewpoint?
Totally insecure. The weaknesses described above provide ample proof to that claim.
On the other hand, we can't really compare today with the situation during WW2. They used machines that were mainly mechanical (eg: Bletchley Park Bombe), while we have computers with powerful CPUs. WW2 cryptanalysts would probably have killed to get their hands on such a futuristic "wunder Maschine" which we use to do simple things like surf the internet.
Nota Bene: If you're interested in the maths behind Enigma, you might want to check on "The Cryptographic Mathematics of Enigma" by Dr. A. Ray Miller, The Center for Cryptologic History – National Security Agency.
The above publication certainly provides more insights related to the internals and maths behind Enigma than the usual NSA "freebee" stuff (PDF) which tends to restrict itself to talking about the history instead of cryptography.
Edit: The last two links are now offline and seem to have been removed from the nsa.gov website. As they are still available via web.archive.org, the links have been updated point there.
There is a full breakdown of key size on a website talking about the “Technical Details of the Enigma Machine”. To sum up:
This is without the plugboard which (with 10 cables) adds a factor of $150,738,274,937,250$ resulting in a 77 bit key.
If the message exceeds $17576$ characters then the key will cycle back to the start. So if there is a longer message you know where to overlap them.
Then there is the issue that letters never map to themselves. The enigma is also weak against a known plaintext in a sequence of cyphertext by a crib attack, as described at a website talking about “The Turing Bombe”.
It looks like with no leakage or errors, Enigma is still secure.
Quoting the Enigma@Home project website:
Enigma@Home is a wrapper between BOINC and Stefan Krah's M4 Project. “The M4 Project is an effort to break 3 original Enigma messages with the help of distributed computing. The signals were intercepted in the North Atlantic in 1942 and are believed to be unbroken.”
Given how the war ended, it would seem all encryption details would have been captured in 1945 - although the Germans might destroyed some of the secret documents about key changing.
I think that short and true messages, for which the rotor positions aren't known, would allow you effectively unbreakable communication. However, in the current digital age, encrypting a large amount of Traffic (eg HTTP) with original Enigma would not be a good idea.
Initially, part of the operation was transmitting three letters twice. It turned out that doing this for about 90 messages with the same settings allowed reverse engineering of the wheels. (I vaguely remember that knowledge of the switchboard was also needed, and that was achieved by non-cryptographic means. My guess is a room cleaner writing down the settings. I don’t know whether in the later more secure operation reverse engineering would have been possible.
In every state, the enigma produces a permutation of 26 letters. A permutation is the product of cycles, for example a->t->f->r->a would be a 4-cycle. So every state produces a pattern of cycle lengths. This pattern is not affected by the switchboard and not affected by rotating the wheels. The pattern can be detected if you have enough messages and when detected, the key space is massively reduced. (Similar to DeCSS where a 40 bit key can be split into a 25 and a 16 bit key).
(All from memory, reality might be different).
