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Assuming people in the forties would have known asymmetric cryptographic methods like RSA, would they have been able to make sensible use of them?

There were no real electronic computers at the time, but the things achieved in Bletchley Park and elsewhere mean that some processes could have been automatised.

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  • $\begingroup$ You probably wouldn't be using anything electronic for regular communication. Those machines, as far as I know, were just used to break encryption. Maybe a more useful question would be "Could one do asymmetric cryptography by hand that could not be broken (in a reasonable time frame) with the computing power available at the time?" $\endgroup$ – Aman Grewal Jul 31 at 17:14
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    $\begingroup$ This question will have opinion-based answers. If it would have been invented then some people would work to build some marvel mechanical devices to easy the calculations. Some mechanical devices like Facit can multiply and divide by zero. So the question, if need do someone able to implement modular square and multiply mechanical machine? Who knows. $\endgroup$ – kelalaka Jul 31 at 17:59
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    $\begingroup$ I do see a problem with RSA / ElGamal / DSA when it comes to generating the key pair. Finding large primes will be extremely tricky. For ECDSA that problem is more or less solved (if you can do ECDH or ECDSA, then you can also create the public key) however, creating the ECDSA domain parameters may be worse than key pair creation. I'm wondering what cryptosystem would allow minimum computation for all operations (domain creation, key pair generation, public key operations and private key operations). $\endgroup$ – Maarten Bodewes Aug 6 at 12:12
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Sure, you can do Diffie-Hellman or RSA by hand if you have to. And you don't need the bit length we use today with powerful adversaries.

I'm not an expert on WW2 computational machines but I won't be surprised if something to do modular exponentiation could be built using WW2 technology given sufficient motivation.

A modern cryptographer in WW2 could make truly unbreakable ciphers. With authentication, key exchange, good randomness, and all the good stuff we have now.

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    $\begingroup$ While one can do Diffie-Hellman or RSA by hand, I want to see some deal with the possible numbers for attacks. For example Wikipedia gives; "In 1845, Izrael Abraham Staffel first exhibited a machine that was able to add, subtract, divide, multiply and obtain a square root". So one can use this Fermat factoring method for RSA. $\endgroup$ – kelalaka Aug 2 at 15:10
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Yes, sure it would be feasible to do modular exponentiation with electro-mechanical machine, that was built especially for this computations. Furthermore, if this technique (of public key) would be known, the operation of "project ultra" was impossible, or much harder, because it was based on the fact that each day, a new symmetric key was randomized and due to the lack of public key technique, was encrypted twice with the previous symmetric key. This means that once the "Bomb" discovered a key, the British could follow this communication channel with Enigma machines only, with no further involvement of "Bomb". If there was a public key technique, than you would need a "Bomb" machine per any German communication channels, operating again each day, means 10's of thousands of Bomb machines, as opposed to less than 10 at the end of the war, which would make things much much harder

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Unfortunately, I cannot find the link, but I have read that some mathematician in the 18th or 19th century already had the idea that functions whose inverse function cannot be calculated with the known methods can be used for what we call "asymmetric encryption" today.

This means that the basic idea of asymmetric encryption was already known before WW I (and of course before WW II). And obviously there were also some functions known whose inverse function could not be calculated at that time. (Otherwise the mathematician would not have had that idea.)

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