Elliptic Curve Cryptography: a gentle introduction

http://andrea.corbellini.name/2015/05/17/elliptic-curve-cryptography-a-gentle-introduction/

by onestone • 11 years ago

238 38 11 years ago

andrea.corbellini.name

valgaze• 11 years ago

If you want to experiment with the graph try adjusting the a/b parameters here: https://www.desmos.com/calculator/3ugvl6yz4i

ufo2• 11 years ago

Needs javascript to read... sigh.

aw3c22• 11 years ago

I was ready to say "well, the animations and formulas need it" but no, those are static images. Meh...

PhantomGremlin• 11 years ago

No it doesn't. At least not with Firefox. Just use

   View --> Page Style --> No Style

But as for the other complaint elsewhere in the comments, about light gray text, unfortunately Firefox can't help with that. Normally such misguided text is countered with

   Preferences --> Content --> Colors --> override ...

but in this case the gray is unfortunately in images and not text.

And it gets even worse for the light gray text. OS X Accessibility has a setting to "Enhance Contrast". But in this case the gray is closer to white than to black, so all that "enhancing" does is make the text even lighter!

Sigh.

Adlai• 11 years ago

The interactive tool[1] needs javascript. Don't try graphing secp256k1.

[1] https://cdn.rawgit.com/andreacorbellini/ecc/0939921/interact...

jgrahamc• 11 years ago

For a slightly less mathematical introduction: https://blog.cloudflare.com/a-relatively-easy-to-understand-...

j2kun2• 11 years ago

But the mathematical ones are the best ones :)

YAYERKA2• 11 years ago

For a slightly more mathematical introduction: https://www.youtube.com/watch?v=t3JzdKE-Fhs along with accompanying slides http://crypto.biu.ac.il/sites/default/files/3rd_BIU_Winter_S... both from ""3rd BIU Winter School on Cryptography: The basics of elliptic curves - Nigel Smart".

ColinWright2• 11 years ago

By my understanding, there's an inaccuracy in that article. It describes a one-way function, but calls it a trapdoor function. I thought that a one-way function is only a trapdoor function if there exists the possibility of reversing it with an extra piece of information. The wikipedia link[0] given seems to confirm that understanding:

    A trapdoor function is a function that is easy
    to compute in one direction, yet difficult to
    compute in the opposite direction (finding its
    inverse) without special information, called the
    "trapdoor".

I know the article is trying to be informal, but that seems a simple thing to get right, and quite important. If I'm wrong I'd welcome being corrected.

[0] http://en.wikipedia.org/wiki/Trapdoor_function

Retr0spectrum• 11 years ago

Wouldn't the "special information" be the private key in this case?

berossm• 11 years ago

It is likely that some of the functions are trap door functions. That is basically what is the theory behind the random number generator that the NSA paid to have set as default in BSAFE library. https://en.wikipedia.org/wiki/Dual_EC_DRBG

YAYERKA2• 11 years ago

In a cryptographic context;

1.12 Definition
A function f from a set X to a set Y is called a one-way function if f(x) is “easy” to compute for all x \in X but for “essentially all” elements y \in Im(f) it is “computationally infeasible” to find any x \in X such that f(x) = y. [0]
1.16 Definition
A trapdoor one-way function is a one-way function f : X -> Y with the additional property that given some extra information (called the trapdoor information) it becomes feasible to find for any given y \in Im(f), an x \in X such that f(x) = y. [0]

[0]; Menezes, A.; Oorschot, P. van; Vanstone, S. (2001). Handbook of Applied Cryptography (5th ed.). CRC Press.

markburns• 11 years ago

I'd advise anyone trying to learn by implementing ECC to be careful using this article as an introduction. It's written as if you can follow along, but there are some omissions (see the comments section).

James D'Angelo has some good videos that go into some of the detail too

https://www.youtube.com/channel/UCgo7FCCPuylVk4luP3JAgVw

markc2• 11 years ago

The Cloudflare ECC introduction is well written and very accessible - however I notice that it omits a very crucial bit of explanation.

In the billards example it's stated that for a given player starting at point A:

"It is easy for him to hit the ball over and over following the rules described above."

arriving at some point B after some number of hits N. However, regarding some other player who knows point A and B

"they cannot determine the number of times the ball was struck to get there without running through the whole game again"

The question of course, is "Why not? Why can't the second player just start at point A and keep hitting the ball until it gets to point B, and count the hits?

The answer is that the first player does not run through the the sequence one hit at a time! Knowing N, you can take a mathematical shortcut from A to B. The simplest of these is known as "double and add" for elliptic curves. This is similar to the "square and multiply" method for fast exponentiation. The shortcut is the critical advantage, because it's the enormous computational complexity of running through the entire game without the shortcut that is the basis of ECC's security.

aw3c22• 11 years ago

Also check out https://events.ccc.de/congress/2014/Fahrplan/events/6369.htm...

Video of the talk: http://media.ccc.de/browse/congress/2014/31c3_-_6369_-_en_-_...

ufo2• 11 years ago

Really liked this version. All the time I was wondering when they would transition from the "toy" clock curve that is easy to understand to the complicated y^3 curves that are mentioned in the OP but it turns out that the easy to understand curve was just as good all along and much easier to implement correctly.

willchang• 11 years ago

There is no design trend more inimical to the function of the web than gray on white text. If only it stopped at gray values of #444444, which is bad enough on old displays or in bright ambient light. The text on this site is #7f8d8c — more white than black — and the strokes on the letters in the equations are quite fine. Yes, it looks nice and clean to have so little black on the page, but the tradeoff is that I have to squint to read it.

freework• 11 years ago

I can't help but to point out the irony of a post complaining about grey text, being... in grey text (referring to the downvotes)

ndesaulniers• 11 years ago

Those of you who know what public-key cryptography

For those of you that don't, might I suggest my: https://nickdesaulniers.github.io/blog/2015/02/22/public-key...

chris_wot3• 11 years ago

You are awesome. Thank you!

S4M• 11 years ago

It's not gonna be the most constructive comment, but I would like to thank the author for writing this article and the poster for submitting it. I worked a bit during my undergraduate studies on the Elliptic Curve method to factor products of large prime numbers, and found it interesting, and sometimes wish I could learn more about that.

tagawa• 11 years ago

For those who prefer audio/video, this was also covered in Security Now episode 374: http://twit.tv/show/security-now/374

tsmarsh• 11 years ago

When I was in college there was a distinct worry about elliptic curves vs rsa. My professors were intuitively worried that discrete logs over elliptic curves may well have a simple solution, unlike factorization which has millenia of research proving that it is hard. They were worried enough that I just assumed that GCHQ had already cracked it. Is that still a worry?

pbsd2• 11 years ago

Such worries have mostly faded by now, though you will see some conservative people still preferring RSA. Keep in mind that the computational study of integer factorization only started seriously in the 70s with CFRAC, making the "milennia" argument somewhat moot.

There is a fascinating article by (among others) Neal Koblitz, one of the inventors of elliptic curve cryptography, which describes the history and development of ECC through the 80s, 90s, and 00s, and some of the worries that developed: https://eprint.iacr.org/2008/390.

chris_wot3• 11 years ago

Can someone help me out here: I'm a bit rusty on my set builder notation... Is the following correct?

The set of natural numbers isn't a group because a group is a set partially defined by { ∀a∃b | a + b = 0 } - or in other words for every element a in the group there exists an element b in the group such that a + b = 0; or to put it yet another way, every element a has an element b that is its arithmetic inverse. As the set of natural numbers are only positive, you can't satisfy this condition.

haversoe• 11 years ago

A group is a set equipped with a binary operator that obeys a few rules. One of the rules is the existence of inverses. If the operator is addition, then your description is correct. That is, every element x of the underlying set has an additive inverse x^-1 also in the set such that x + x^-1 = 0. Obviously, zero needs to be in the set as well and in some constructions of the natural numbers it is not.

chris_wot3• 11 years ago

Thank you!

master-lincoln• 11 years ago

I don't like how 0 is used as three different symbols in the beginning: the number, the point of infinity and the identity element. I know 0 is a common symbol for each of the three cases respectively but in the same context it feels rather confusing. Is that a common thing to do?

geogriffin• 11 years ago

the number zero is always defined as being an additive identity element, and in these groups 0 is the point at infinity.

jordigh• 11 years ago

Yes, it's a very common thing to use the same symbol to mean different things in mathematics.

Elliptic Curve Cryptography: a gentle introduction

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