crypto/rsa: doc: reword "coprime to" in doc comments

[as]

I found this paragraph in the crypto/rsa docs:

https://github.com/golang/go/blob/master/src/crypto/rsa/rsa.go#L187-L191

	// Check that de ≡ 1 mod p-1, for each prime.
	// This implies that e is coprime to each p-1 as e has a multiplicative
	// inverse. Therefore e is coprime to lcm(p-1,q-1,r-1,...) =
	// exponent(ℤ/nℤ). It also implies that a^de ≡ a mod p as a^(p-1) ≡ 1
	// mod p. Thus a^de ≡ a mod n for all a coprime to n, as required.

It is a mistake to say x is coprime to y, because co and to convey the same thing. A number x, y can be coprime, but we say x is prime to y.

This is explicitly stated by Stewart in Galios Theory 3rd Edition (ISBN 1-58488-393-6), in Definition 3.14 on the topic of coprimes.

[jimmyfrasche]

Data point: I've always heard and said "x coprime to y". I don't recall ever hearing "x prime to y".

cc: @FiloSottile

[as]

@jimmyfrasche The more familiar utterance is "relatively prime to y", although my source text does not mention it

[kurahaupo]

I've usually heard it said as “X is co-prime with Y”, which follows from the general pattern in English of “X and Y are Z” being often interchangeable with “X is Z with Y”.

The usage queried here follows by combining that with another general pattern in 21st century English: to replace “with” with “to” in most common phrase patterns.

[smasher164]

If x and y are coprime, we also say that x is coprime to y, or that y is coprime to x.

Bourbaki, N. (2003). Algebra II: Chapters 4-7. Berlin: Springer-Verlag. A. VI. 14

I think it's fine the way it's currently written.

[as]

Stewart's text was published in 2015, and authors under the Bourbaki pen name publish their books in French. Is your 2003 source a translation of their original work?

[smasher164]

It is, yes, of their work in 1990, but that is only example. A google scholar search for "is prime to" AND "RSA" leads to 399 results, half of which oddly don't actually have the phrase "is prime to". However, a similar search for "is coprime to" AND "RSA" shows nearly 785 hits use this as standard terminology.

Taking a look at the original RSA paper, the terminology used is actually "x is relatively prime to y". I suspect that "x is coprime to y" became popular because of its terseness, as well as its ability to clearly communicate that these values are not individually prime. If we were to change anything, I'd gravitate toward "is relatively prime to".

[as]

I'd gravitate toward "is relatively prime to".

This seems like an agreeable compromise.