Cryptography Stack Exchange
Cryptography Stack Exchange

Questions tagged [elliptic-curves]

Elliptic curves are algebraic-geometric structures with applications in cryptography. Such a curve consists of the set of solutions to a cubic equation over a finite field equipped with a group operation. Questions relating to elliptic curves and derived algorithms should use this tag and might also consider more specific tags such as discrete-logarithm and ecdsa.

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Why do the elliptic curves recommended by NIST use 521 bits rather than 512?

Wikipedia says in reference to the elliptic curves officially recommended by NIST in FIPS 186-3: Five prime fields for certain primes p of sizes 192, 224, 256, 384, and 521 bits. For each of the prime fields, one elliptic curve is recommended. The…
Zack Elan
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Why would anyone use an elliptic curve with a cofactor > 1?

In cryptography, an elliptic curve is a group based on a finite field $GF(p^k)$; this group has $n$ elements on it, and we work on a prime-sized subgroup of size $q$. We denote the value $h = n/q$ as the cofactor of the curve. My question is: why…
poncho
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How does ECC go from decimals to integers?

I realise that elliptical curves are tricky, but there's one aspect that no one seems to explain. I've looked, and it's towards the beginning. This is the traverse over a red curve:- This is only three steps (n=3). For cryptography n must be…
Paul Uszak
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Does the elliptic curve (EC) cryptosystem outperform RSA and DL cryptosystems?

Throughout the literature, it is stated that EC cryptosystems outperform RSA and Discrete logarithm cryptosystems, but I cannot understand how ECC would be more efficient than RSA and DL in terms of computation and storage. Is there any pragmatic…
user4665
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How does one calculate the scalar multiplication on elliptic curves?

I found this example online: In the elliptic curve group defined by $$y^2 = x^3 + 9x + 17 \quad \text{over } \mathbb{F}_{23},$$ what is the discrete logarithm $k$ of $Q = (4,5)$ to the base $P = (16,5)$? One (naï­ve) way to find k is to…
Keith Lau Si Keit
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Why Elliptic Curves?

What is the benefit of using elliptic curves over the standard finite field, when the cyclic subgroup we consider of the EC's solution group is just isomorphic to some integer residue class of prime order? Is it because the group operation is more…
dylan7
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When do ECC patents end?

As the topic says, since when can ECC cryptography be freely used? Is it not widely used because of patents? There is no alternative to it on embedded devices and smart cards. Just to mention: I am not from the USA. I'm a simple user who wants ECC…
Smit Johnth
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When adding two points on an elliptic curve, why flip over the x-axis?

Every introduction to elliptic curves that I've read hasn't explained this. If you have two points P and Q on an elliptic curve, to find P+Q, you draw a straight line through the points, find the third point of intersection with the elliptic curve.…
Kevin
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Why are Jacobian Coordinates used?

I couldn't find this explained in another question, but is there an actual reason as to why Jacobian coordinates are used for elliptic curves? Do they provide some sort of advantage in terms of performance? Or are they easier to use for security…
Lev Knoblock
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How are points on an elliptic curve discretized?

I'm a working programmer (read: a person without a maths degree) trying to get a better grasp on elliptic curves specifically in the context of elliptic curve cryptography (though to be clear, this is for personal development — I'm in no way trying…
QuartzCrystal
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What is an elliptic curve cofactor?

As the title says, I have some doubts about the term "cofactor" used to describe elliptic curves. AFAIK, it's a factor of the curve order, but why is it explicitly specified in some parameter lists then? How does it apply to the curve point addition…
Mark
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Can elliptic curve cryptography encrypt with public key and decrypt with private key like RSA?

I know that RSA can be used for both, encryption and signature. What about EC? I know about ECDSA/EdDSA, but to my knowledge it can only be used to sign. I also know about ECDH, but it is a key agreement protocol. Is there some elliptic curve based…
Eric
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Why can an elliptic curve private key be 1?

I often see in papers (e.g. this one) that for an elliptic curve with generator point $G$ and order $n$ the private key $d$ can take on any integer value in the range $[1, n)$. When $d = 1$ the corresponding public key $Q = dG$ is of course then…
puzzlepalace
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Point halving on elliptic curves of even order

I am trying to understand how point halving on elliptic curves of even order works. Specifically: suppose $g$ is an elliptic curve, and $G$ is a generator point on this curve. The order of group generated by $G$ is even and equal to $q$. Suppose…
irakliy
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EC curve selection

Say for instance, this webservice offers the following curves. sect283k1 sect283r1 sect409k1 sect409r1 sect571k1 sect571r1 secp256k1 prime256v1 secp384r1 secp521r1 brainpoolP256r1…
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