Python cryptography, Rust, and Gentoo [LWN.net]

Python cryptography, Rust, and Gentoo

Posted Feb 13, 2021 1:39 UTC (Sat) by Wol (subscriber, #4433)
In reply to: Python cryptography, Rust, and Gentoo by NYKevin
Parent article: Python cryptography, Rust, and Gentoo

> 3. UB that should always crash - Mostly, this is just "dereferencing NULL, dividing by zero, and anything else that everyone agrees should always immediately trap,"

And here we have to disagree - computers are supposed to do maths, and division by zero is a common mathematical operation. The result is (scalar) infinity, I believe, and it's actually absolutely fundamental to that branch of mathematics known as calculus.

(One of the problems people have with infinity(s) is that there are so many, and you can't mix them ... :-)

One of my early projects that I remember involved a lot of Pythagorus. The problem was, the three vertices of my triangle could easily lie on a straight line, which would result (as far as the maths was concerned) in a "divide by divide-by-zero". To which the answer is zero. As far as the program was concerned, though, it did result in a crash, resulting in a load of extra code to trap the fact that computers can't do maths properly :-)

I don't know whether the language people are doing this, but imho they should get rid of both implementation-specific behaviour, and undefined behaviour. Let's take the example of shifting a negative amount. imho the principle of least surprise says that a negative left shift is a right shift, so if you explicitly ask for the new standard you get the defined behaviour. Unless you also ask explicitly for the old behaviour. If you don't ask for anything it remains implementation-specific (until the compiler default advances to the new standard :-) And fix undefined behaviour the same way - that should only be allowed when asking for something non-sensical :-)

They had this exact problem with FOR/NExT loops in 1977 :-) FORTRAN did the test at the end, so all loops executed at least once, while Fortran77 did it at the start so loops could possibly not execute at all. So we had switches to force either new or old behaviour.

Cheers,
Wol

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Python cryptography, Rust, and Gentoo

Posted Feb 13, 2021 9:02 UTC (Sat) by mkbosmans (subscriber, #65556) [Link] (4 responses)

> And here we have to disagree - computers are supposed to do maths, and division by zero is a common mathematical operation.
> The result is (scalar) infinity, I believe, and it's actually absolutely fundamental to that branch of mathematics known as calculus.

That is not the case at all.
While you can say: lim _x→0 n/x = inf, it does not follow that n/0 = inf.

And as for the more general point, calculus deals with real numbers for the most part. Computers operate on floating point and integer numbers. Operations that make sense in one domain don't necessarily translate 1:1 to another.

Python cryptography, Rust, and Gentoo

Posted Feb 13, 2021 10:09 UTC (Sat) by Wol (subscriber, #4433) [Link] (3 responses)

> That is not the case at all.
> While you can say: lim x→0 n/x = inf, it does not follow that n/0 = inf.

But isn't that what the MATHS MEANS? It doesn't follow that x will *reach* 0, but if it does, then n/x *must* equal infinity. (Quite often, x=0 is illegal in the problem domain.)

> And as for the more general point, calculus deals with real numbers for the most part. Computers operate on floating point and integer numbers. Operations that make sense in one domain don't necessarily translate 1:1 to another.

Principle of least surprise. Yes, floating point is a quantum operation, while reals aren't, but given that (I believe) the IEEE definition of floating point includes both NaN and inf, it would be nice if computers actually used them - I believe some popular computers did 40 years ago (DEC Vax), and I guess it's the ubiquity of x86 that killed it :-(

And the whole point of fp is to imitate real. Again, principle of least surprise, the fp model should not crash when fed something that is valid in the real domain. It should get as close as possible.

People are too eager to accept that "digital is better" "because it's maths", and ignore the fact that it's just a model. And people find it hard to challenge a mathematical model, even when it's blatantly wrong, "because the maths says so".

Cheers,
Wol

Python cryptography, Rust, and Gentoo

Posted Feb 13, 2021 11:38 UTC (Sat) by Jonno (guest, #49613) [Link]

> While you can say: lim x→0 n/x = inf, it does not follow that n/0 = inf.

No, you can't say that. For n∈ℝ⁺, lim (x⭢0)⁺ (n/x) = ∞, but lim (x⭢0)⁻ (n/0) = -∞, so lim x⭢0 (n/0) does not exist. [For n∈ℝ⁻, lim (x⭢0)⁺ (n/x) = -∞ and lim (x⭢0)⁻ (n/x) = ∞, so lim (x⭢0) does not exist either; but for n∈{0}, lim (x⭢0)⁺ (n/x) = 0 and lim (x⭢0)⁻ (n/x) = 0, and so lim (x⭢0) (n/x) = 0].

> But isn't that what the MATHS MEANS? It doesn't follow that x will *reach* 0, but if it does, then n/x *must* equal infinity. (Quite often, x=0 is illegal in the problem domain.)

No, the maths says that the domain of the divisor does not include zero; that the closer to zero a positive divisor gets, the closer to positive infinity the value gets; and that the closer to zero a negative divisor gets, the closer to negative infinity the value gets.

Python cryptography, Rust, and Gentoo

Posted Feb 13, 2021 13:15 UTC (Sat) by mpr22 (subscriber, #60784) [Link]

x86 floating point is (hardware defects aside) IEEE-754, floating point division by 0.0 is defined in C, and if you compile:

#include <stdio.h> int main() { float f = 1.0f/0.0f; printf("%f\n", f); return 0; }

with gcc or clang and link against glibc, it prints "inf".

Integer division by 0, on the other hand, is undefined under finite-width two's complement (or unsigned) arithmetic.

Python cryptography, Rust, and Gentoo

Posted Feb 14, 2021 2:47 UTC (Sun) by NYKevin (subscriber, #129325) [Link]

Infinity is neither an integer nor a real number (when both terms are defined in the mathematical sense rather than the computational sense). The real numbers observe something called the "Archimedean property," which states that there are no infinities or infinitesimals (except that zero is infinitely smaller than all non-zero values).

Why do real numbers have this limitation? Well, the blunt fact is, there's only one totally ordered metrically complete field, and it's the real numbers.[1] If you want to introduce an infinity, you have to give up one of the following:

1. The field axioms (which, broadly speaking, say that you can add, subtract, multiply, and divide real numbers, and these operations behave in sensible and familiar ways).
2. The total ordering of the reals (i.e. for any two reals a and b, either a > b, a < b, or a = b, where = is given its ordinary interpretation of "are literally the same mathematical object" rather than something which the ordering is allowed to define).
3. Two additional axioms about how the ordering interacts with the field operations (basically, you can always add the same number to both sides of an inequality without invalidating it, and the product of two positive numbers is always positive).
4. The reals are Dedekind-complete (in simple terms, "you can take limits" - in more precise terms, every non-empty subset of the reals that has an upper bound, has a least upper bound).

For example, IEEE 754:

1. Everything is non-associative, which is not allowed under the field axioms. Also, NaN and ±inf don't have additive inverses.
2. Since 1.0 / 0.0 != 1.0 / -0.0, we cannot have 0.0 and -0.0 be "the same value" (because you get different answers when you try to use them in the same expression). Neither number is greater than the other according to IEEE 754, and so they violate total ordering. Also, all of the NaNs violate total ordering, too.
3. There are cases for which x < y, but x + z == y == y + z (because x is the largest value with exponent n and y is the smallest value with exponent n+1). Also, you can trivially break this with ±inf.
4. Almost satisfied: The set of negative floats has two upper bounds which are incomparable (0.0 and -0.0), so we cannot say which is the "least" upper bound. But I'm pretty sure this is the only counterexample (ignoring trivial alterations such as "negative floats greater than -1.0," etc.) because I can't think of a way to construct a counterexample out of NaN or inf.

Or the extended real numbers, which IEEE 754 is intended to mimic:

1. inf - inf is not defined (rather than giving an NaN), which is not allowed under the field axioms. Also, ±inf don't have additive inverses.
2. Satisfied if you assume that -inf < all reals < inf.
3. Not satisfied because, for finite numbers x and y with x < y, x + inf = y + inf = inf.
4. The extended reals are compact, which in this context is an even stronger property than completeness.

Or the hyperreal numbers, which are explicitly designed to "follow all of the usual rules" (for use in nonstandard analysis):

1. Satisfied by the transfer principle.
2. Satisfied by the transfer principle.
3. Satisfied by the transfer principle.
4. Not satisfied: Consider the set of infinitesimals. This is clearly bounded, but it cannot have a *least* upper bound, or else you could derive contradictions by doubling this least upper bound (which must give you a non-infinitesimal) and reasoning about the relationship between the resulting number and the set of infinitesimals.

TL;DR: If you like doing calculus etc. in the usual way, then you can't have infinities or infinitesimals.

[1]: Every totally ordered metrically complete field is isomorphic to the real numbers. So they're all, effectively, "the same field" with different names for their elements. We need this caveat because, if you really wanted to, you could just start calling 2 ** 128 "infinity." But it would still be 2 ** 128. You could still multiply two by itself 128 times to get to your "infinity." "A rose by any other name" and all that.

Python cryptography, Rust, and Gentoo

Posted Feb 13, 2021 15:59 UTC (Sat) by mafrasi2 (guest, #144830) [Link] (7 responses)

> And here we have to disagree - computers are supposed to do maths, and division by zero is a common mathematical operation. The result is (scalar) infinity, I believe, and it's actually absolutely fundamental to that branch of mathematics known as calculus.

> (One of the problems people have with infinity(s) is that there are so many, and you can't mix them ... :-)

Division by zero is *not* a common mathematical operation. It is literally undefined in mathematics as well.

In fact, it is absolutely fundamental that it is undefined, because otherwise you could do for any number x

x / 0 = infinity
x = infinity * 0

which would mean that any number is equal to any other number (because they all equal infinity * 0 and equality is transitive).

Python cryptography, Rust, and Gentoo

Posted Feb 13, 2021 23:57 UTC (Sat) by Wol (subscriber, #4433) [Link] (6 responses)

Except that anything multiplied by 0 is zero.

So we have infinity *0 = 5*0 = 4*0 =3*0 = 2*0 = 1*0 =0*0.

If we divide each term by 0, does that mean infinity = 5 = 4 = 3 = 2 = 1 = 0?

I think we've fallen foul of Godel's incompleteness theorem. In order to make the maths work, we need special rules outside of the maths like "divide by zero, you get infinity" and "divde by infinity, you get zero". And a whole lot of physics depends on infinities. I can't give you any examples (or maybe I can), but there are various different types such that quite often infinity != infinity, and the physics doesn't work. And the "is it 10 or 11 dimensions" model of space-time works, I believe, because it just happens to be true that infinity does actually equal infinity.

Infinity and zero are special cases, required by Godel, that are needed to make everything else work. Take that example I gave of calculating the sides of a triangle - as soon as we accept that "divide by divide-by-zero equals 0" I can use THE SAME maths on any two points in a cartesian system to calculate the distance between them. Basically I try and construct a right angle triange and calculate the hypotenuse, and if the triangle collapses into a line THE SAME maths still works. And it makes sense that it works...

Cheers,
Wol

Python cryptography, Rust, and Gentoo

Posted Feb 14, 2021 1:28 UTC (Sun) by mathstuf (subscriber, #69389) [Link] (5 responses)

> So we have infinity *0 = 5*0 = 4*0 =3*0 = 2*0 = 1*0 =0*0.

inf * 0 is an indeterminite form. It isn't zero, it isn't inf, it isn't a number. Your logic just breaks down here.

> I think we've fallen foul of Godel's incompleteness theorem.

Umm, no. This is way before Gödel gets involved.

> In order to make the maths work, we need special rules outside of the maths like "divide by zero, you get infinity" and "divde by infinity, you get zero".

No, these rules don't exist (in normal mathematics, see later). They may "make sense" in specific instances, but they are nonsense if you try to extrapolate from them. Dividing by zero is not an operation you can do. It isn't inf, nan, or any other "thing", it just can't be done (at least in the axiomatic framework generally used; IEEE is notably lacking in axioms, so sure inf is fine there). I'm sure one could make an algebra where division by zero "makes sense" (cf. modular algebra or surreal numbers for other number systems; surreal *might* have division by zero, but it is…weird), but it might not be as useful as the algebra we use all the time.

> And a whole lot of physics depends on infinities.

I think you mean infinite series or infinitesimals, not infinities.

> I believe, because it just happens to be true that infinity does actually equal infinity.

Maybe you're thinking of the continuum hypothesis? Though I don't think string theory cares about it in particular (its truth is independent of ZF or ZFC). Though I don't know for sure in that specific instance.

> Infinity and zero are special cases, required by Godel,

I feel like you're not understanding Gödel. Gödel states that there are truths that are unprovable in any given proof system that is consistent. Or you can have all truths, but then you gain all falsities as well without the power to tell the difference. There's nothing in it about infinity or zero (as applied to number theory). Those existed before Gödel came along and are fine. I recommend the book Gödel's Proof by Nagel and Newman which is what finally turned the light bulb on for me (after not getting it in Gödel, Escher, Bach by Hofstadter and another reference I can't remember).

Python cryptography, Rust, and Gentoo

Posted Feb 14, 2021 10:06 UTC (Sun) by Wol (subscriber, #4433) [Link] (2 responses)

As I understand Godel, a simple way to put it is "you cannot use a system to prove itself correct". So it's easy to prove boolean logic correct, AS LONG AS you don't restrict the proof to using only boolean logic. It's easy to prove number theory correct AS LONG AS you don't restrict tthe proof to using only number theory. That's why we can't prove logic correct, because we have nothing else to throw into the mix.

So I have no qualms about throwing that infinity stuff into the proof, because otherwise you can't class zero as a number, because it behaves completely differently to all the other numbers. "My logic breaks down". Yes, because my logic (as per Godel) MUST be either incomplete, or inconsistent. Without that rule, it's inconsistent. With that rule it's incomplete. Pick one ... I've gone for consistency.

Cheers,
Wol

Python cryptography, Rust, and Gentoo

Posted Feb 14, 2021 13:08 UTC (Sun) by mathstuf (subscriber, #69389) [Link] (1 responses)

> As I understand Godel, a simple way to put it is "you cannot use a system to prove itself correct".

There are two parts.

The first is that any sufficiently powerful[1] system of arithmetic is incomplete. In this sense it means that there are statements one can make in the system for which no proof exists (of either its truth or falsity).

The second is that in such a system, the consistency of the system itself is one such statement.

It makes no such claim as to which statement is required.

> That's why we can't prove logic correct, because we have nothing else to throw into the mix.

Sure, but *that* system is also not provably correct. So what have you gained? You (claim to have) jumped one rung up a countably infinite ladder among a countably infinite selection of such ladders. Yay? :)

> So I have no qualms about throwing that infinity stuff into the proof, because otherwise you can't class zero as a number, because it behaves completely differently to all the other numbers.

Zero is a number. It works just fine. Division has a singularity at its value, but all kinds of functions have singularities. Do we need something else for tan(π/2)? Why not extend to the complex numbers with sqrt(-1) while we're at it? Quaternions? Octonions? Sedenions? Each of these is a separate algebra, an extension of algebra over the reals. We don't use them in general because we don't need the additional power they offer in day-to-day uses.

> Yes, because my logic (as per Godel) MUST be either incomplete, or inconsistent. Without that rule, it's inconsistent. With that rule it's incomplete. Pick one ... I've gone for consistency.

You're using the wrong definition of "consistency". It isn't consistent as in "all values must be able to take places of all other values in all expressions".[2] It is consistent as in "there are no contradictions between provable statements" which is *way* more important in (useful) mathematics.

[1] Peano arithmetic is sufficiently powerful. Arithmetic with just the natural numbers, addition, and multiplication, I believe, is not.
[2] You're still "inconsistent" in this sense about the square root of negative numbers for example. Why not toss those in? Why stop at trying to make division "consistent" in this sense when you're leaving out the trigonometric functions, square root, and the other infinite singularity-containing or domain-limited functions alone?

Python cryptography, Rust, and Gentoo

Posted Feb 14, 2021 16:34 UTC (Sun) by nix (subscriber, #2304) [Link]

> Each of these is a separate algebra, an extension of algebra over the reals. We don't use them in general because we don't need the additional power they offer in day-to-day uses.

... and because they gain annoying limitations (lose a useful property of the reals) with every such extension, and by the time you get to the sedenions there's not really very many useful properties they have left (associativity is more or less it).

Python cryptography, Rust, and Gentoo

Posted Feb 15, 2021 7:17 UTC (Mon) by NYKevin (subscriber, #129325) [Link] (1 responses)

> I feel like you're not understanding Gödel.

This is nothing to be ashamed of, by the way. I took an entire college course just focusing on the incompleteness theorems, and I still only have a very loose ability to follow their basic form. The incompleteness theorems are very, *very* hairy math. You cannot simply skim a couple of Wikipedia articles and expect to understand Gödel.

If you insist on trying to figure out what Gödel was saying without spending multiple years of your life studying the surrounding mathematics, then I would suggest starting out with Gödel Escher Bach. Yes, that's a very thick book, no, you should not skim it. The main advantage of GEB is that it actually does explain why and how completeness breaks down under arithmetic, using a real (if rather awkward) implementation of PA. For this reason, it is not an easy read, but it's better than an introductory model theory textbook.

Python cryptography, Rust, and Gentoo

Posted Feb 15, 2021 22:46 UTC (Mon) by rgmoore (✭ supporter ✭, #75) [Link]

GEB is not an easy read, but it is probably as easy and fun a read as any book that makes a serious pretense of explaining Gödel's Incompleteness Theorem is likely to be.