What are elliptic curve, why are they so important?

[cerulean vale]

Hello, just came across the term elliptic curve. From what I've understood, these are just ellipse but they are important when it comes to the world of cryptography.

Can someone explain why please, what do they have which are so special?

[cerulean vale]

anyone pls

[neat nest]

No, elliptic curves are not ellipses. Despite the name, they are defined by a different kind of equation — typically of the form
    y^2=x^3+ax+b
for constants a and b, under certain conditions.

They are important for cryptography because these curves allow a nontrivial group structure on them (see what a group structure in algebra means). Moreover, in many cases the discrete logarithm problem is hard to solve for these groups and the addition law in that group can be implemented with fast algorithms.

And in general, the rich algebraic structure of elliptic curves plays a key role in modern number theory and has led to deep results, such as in the proof of Fermat’s Last Theorem.

[cerulean vale]

Oh ok, what's the deal with prime numbers and their factorisation? It's difficult to factorise them?

[neat nest]

yes, if a composite number is huge, like 10^200, and has no small divisors, then it's hard to factorize it.

[cerulean vale]

Oh ok I see, thanks !

[cerulean vale]

by the way, an ellipse and an elliptic curve isn't the same thing?

[neat nest]

Yes

[cerulean vale]

yep, one last thing, even for a computer it is difficult to find the prime numbers? is there a reason for that? I mean after all, computers should be "faster" than humans ?

[neat nest]

Checking if some given number is prime or composite is easy for the computers, even for these huge numbers. But if the number is composite and huge, finding it's factors is hard even with most powerful computers.

[cerulean vale]

oh ok I see, thanks !

[hardy hazel]

elliptic curve isnt ellipse

name sucks, absolute total history accident

anyways

its just

"super duper specific type smooth, loopy equation"

like y² = x³ + ax + b

anyways

have you ever heard of scalar multiplication

(multiplying point G by number k)

[hardy hazel]

ok but before going deeper into scalar multiplation

ima explain u stuff abt dot jumping

• imagine we have 2 points on the snake line (elliptic curve)
• where does it hit the curve? thats point C (Actually, you flip it over the x-axis for technical reasons, but ignore that)
• crucially: doing ts dot jumping (point addition) is literally computationally easy

for scalar multiplication:

• lets imagine if we do this dot jumping over and over
• start at a specific generator (point G)
• do G+G = 2G
• then 2G+G = 3G
• then 3G+G = 4G and so on
  = scalar multiplication ✨

which apparently

when going forward (lets say if i tell u k=5) its relatively fast to compute kG (5 jumps from G)

but the absolute brutal thing is that

if you try to go backwards, in simple words if i show you the result only (kG) and ask what number (k) did i use to get here from G? this is the Elliptic Curve Discrete Logarithm Problem (ECDLP). for huge ahh curves this is equivalent to trying to find a specific grain of sand on all the beaches on Earth LOL. even top insane ahh computers give up on these calculations n go back crying 🙏😭

now heres the actual question

"Why not use regular math like RSA?"

answer to this is

we have other trapdoors such as factoring huge ass primes in RSA but this is why its important in crypto (elliptic curve)

• Smaller Keys = More Speed, Less Space: To get security equivalent to a massive 3072-bit RSA key, you only need a 256-bit elliptic curve key. Think phone vs. brick. Your phone can do ECC crypto fast without draining the battery. Websites load quicker.
• Same Security, Less Hassle: Breaking ECC is currently way harder than factoring big numbers for the same computational cost. So you get tougher security with less math overhead.

[cerulean vale]

oh ok I see interesting, I only knew RSA thing, didn't know the ECC cryptocraphy