Mathematics desk
< July 13	<< Jun \| July \| Aug >>	Current desk >

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

July 14[edit]

What is the exact probability of secret technology?[edit]

Is there actually any form of serious scientific work that addresses the likelihood that there is any secret technology? I mean something like a work in statistics. 2A02:908:424:9D60:6D5D:2601:E025:C28D (talk) 16:19, 14 July 2022 (UTC)[reply]

If one defines 'a secret' as 'data not available, intentionally', as would seem a reasonable starting point in this context, then I think we can conclusively state that the probability lies somewhere between 0 and 1. Exact answers to vague questions are rarely forthcoming in statistics (or in most other contexts), and when asking a question that implies that data wouldn't be available to answer it even if better defined, only optimism beyond any reasonable bounds would justify expecting any sort of sensible response at all. TLDR: you can't do statistics on things you don't know about... AndyTheGrump (talk) 16:39, 14 July 2022 (UTC)[reply]

(edit conflict) Some corporations do not publish information about some of the key technologies they have developed. They choose not to apply for a patent, but to protect their inventions by keeping them a trade secret, much like the Coca-Cola corporation with its secret formula, kept in a vault at the World of Coca-Cola museum in Atlanta. Much military technology, such as state-of-the-art stealth technology, is also a closely guarded secret. So it is a 100% certainty that secret technology exists. --Lambiam 16:42, 14 July 2022 (UTC)[reply]

One can look at the probability that a conspiracy remains secret over time, that is, model the people, not the technology. That still requires many assumptions, however. See On the Viability of Conspiratorial Beliefs. --{{u|Mark viking}} {Talk} 16:54, 14 July 2022 (UTC)[reply]

You might read up on the difference between probability and Knightian uncertainty. By secret technology, if you mean something like Star Trek aliens, I'd say you're pretty stuck. If you're in the WW2 German Army, the Allied cryptanalysis of the Enigma is an unknown technology, but let's say you hypothesize its existence and you want to estimate the probability. You could look at the observed progress of some past enemy actions like the North Africa Campaign where the Allied movements were in fact informed by decoding Rommel's encrypted radio traffic. Let the null hypothesis H0 be that the Allies have not broken Enigma. In that case, their movements will have been guided by normal military doctrine, which (consistent with game theory that would be formalized later) will involve a number of random choices. Still, you can find probability distributions for what they would do without the benefit of Enigma intelligence, vs. what they would do with that benefit. You can then do some Bayesian inference or use something like the Kolmogorov-Smirnov test, and from there get a probability estimate that Enigma is broken.

The novel Cryptonomicon doesn't go into such detail, but part of its plot is that a German army mathematician does in fact figure out that Enigma is broken. He informs his superiors, and describes afterwards what happened:

"I found weaknesses everywhere," von Hacklheber says. "Most codes were designed by dilettantes and amateurs with no grasp of the underlying mathematics. It is really quite pitiable."

"Including the Enigma?" Bischoff asks.

"Don’t even talk to me of that shit," von Hacklheber says. "I dispensed with it almost immediately."

"What do you mean, dispensed with it?" Root asks.

"Proved that it was shit," von Hacklheber says.

"But the entire Wehrmacht still uses it," Bischoff says.

Von Hacklheber shrugs and looks at the burning tip of his cigarette. "You expect them to throw all those machines away because one mathematician writes a paper?"

One lesson from this is that a secret weapon becomes less secret every time you use it. Coventry_Blitz#Coventry_and_Ultra describes a legend (i.e. probably false) based on that principle. 2602:24A:DE47:B8E0:1B43:29FD:A863:33CA (talk) 02:37, 15 July 2022 (UTC)[reply]

I don't think the Enigma machine itself was assumed secret; it was designed so that a machine could fall into the hands of the Allies (which it did), without the individual messages being insecure. There was a "key" that had to be entered into the machine and those were changed every day. As far as I know none of the actual keys were ever acquired by the Allies until after the messages were sent. My understanding is that the keys were useful for refining decryption techniques, but none produced any current information. What was a secret was the Allied technology for decoding the messages. I don't think anyone on the Axis side would have conceived of someone creating a machine with the computing power to crack the individual keys in a reasonable about of time (say, before the war ended). The same principle exists today; the technology behind Pretty Good Privacy is public domain, but as long as you keep keys secure then messages are secure. But this is under the assumption that no one will devise technology which will break keys. If that technology exists, then it must be such a closely guarded secret that only a few people know about it. --RDBury (talk) 10:51, 15 July 2022 (UTC)[reply]

For exactly this reason, the Enigma machine in its original form was sold commercially: the idea was that businesses would buy it to protect their communications from corporate spies. That's why it had a catchy name instead of something obscure like most military secrets (e.g. tank, Manhattan Project). The German military improved the machine when they started using it themselves, though. --174.95.81.219 (talk) 03:15, 20 July 2022 (UTC)[reply]

I wasn't trying to be historically accurate further up, but just giving a thought experiment about inferring the existence of unknown technology. The novel I quoted wasn't accurate at all, But, the Enigma was cracked by abstract algebra and the computation involved didn't search all possible keys or anything like that. Rejewski's analysis using permutation groups is sometimes called "the theorem that won World War II". It is reproduced in an appendix of Władysław Kozaczuk's book "Enigma" (hardcover edition only, I think). 2602:24A:DE47:B8E0:1B43:29FD:A863:33CA (talk) 16:36, 15 July 2022 (UTC)[reply]