congratulations, you've performed an autopsy on a joke

d=4 is not a valid RSA private key. When p and q are prime λ(n) = lcm(p − 1, q − 1) is always even. e must be coprime with λ(n) so it's odd. de ≡ 1 (mod λ(n)) which means that de is 1 + an even number, so d must also always be odd.

I assume you’re referring to [1]? If so, I just did after seeing your comment. I couldn’t help but chuckle at the demonstration of how easy it would be for someone to obtain the private keys themselves using an LLM, even though it’s easy to crack without one….

[1] https://bugzilla.mozilla.org/show_bug.cgi?id=2083449

Yeah, except in total there were eight different weak keys, all less than 256. The odds of randomly generating those keys (all so small) is so astronomically low that we know for sure their keygen process was messed up.

Using RSA as an example, the prime factors are supposed to have the same bit length (plus or minus 1 for carryover) for a similar reason. If the smaller factor has 1 fewer bit than the larger factor, it will be discovered twice as quickly by brute force, and 4x faster if it has 2 fewer bits, 8x faster if it has 3 fewer bits, etc. You could technically create a 4096 bit RSA key where one of the factors is 3 and the other is some other giant prime, but it would be immediately discovered by a naive brute force attack. The algorithm itself isn't concerned with the bit-length difference and operates on the inputs opaquely. The key generator is responsible for selecting prime factors that cannot be easily derived from the algorithm's output.

Sure, but the probability of getting "4" as opposed to some number that might as as well be equivalent to latter as far as a causal reader is concerned is insanely small.

There's only a ~1 in 17 chance that they'd all be below 256 if 384 was the limit. My money is on accidentally generating a 1-byte random number.

That's a 4% chance. (2/3)^8