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 ...

learn more… | top users | synonyms (1)

2
votes
1answer
99 views

What does signed fixed window method mean in ECC?

I am studying (sliding) window method in Elliptic Curve Cryptography (ECC) but I am confused by the term, signed fixed window method. By the way term is used in a research paper and not in the book ...
2
votes
1answer
201 views

Is the term “Elliptic Curve Discrete Logarithm Problem” a misnomer?

I have just started studying Elliptic Curve Cryptography, and I have this doubt. In ECC the group operation is addition (and not multiplication). So, why is ECDLP stated as a variation of the discrete ...
2
votes
2answers
232 views

What is the fastest elliptic curve operation f(P) in affine coordinates such that f^n(P)=P only if n is large?

I'm working with the affine representations of points of the Secp256k1 elliptic curve (from Bitcoin). I've read many papers that show that computing some functions, like $f(P)=3P$ can be computed ...
2
votes
2answers
140 views

In ECC, how do I prove that point addition is commutative?

I am studying elliptic curve cryptography and this question is related to the commutative property of point addition operation. Point addition $P_3(x_3,y_3)$ of two points $P_1(x_1, y_1)$ and ...
2
votes
1answer
80 views

What are the differences between the elliptic curve equations?

I think we're all aware of the "classical" Weierstrass (short?) elliptic curve equation: $y^2\equiv x^3 + ax +b \pmod p$. Well known examples of these curves include the NIST's and Brainpool ones. ...
2
votes
1answer
73 views

Doubling a point on an elliptic curve

I'm working with the elliptic curve $\mathcal{E} : y^2 = x^3 + 11x^2 + 17x + 25$ over $(\mathbb{Z}_{31},+,\cdot)$ and am trying to double $P=[2,7]$. Following the instructions here, I'm doing the ...
2
votes
1answer
193 views

Elliptic curve cryptography G*G

I understand how ECC works for the point multiplications and stuff. All we normally do is to multiply a scalar number (lets say d as a private key) with the base point generator ...
2
votes
2answers
279 views

Solving Quadratic equations in Galois Field (2^163)

Hello I am working on implementing a message to elliptic curve point mapping hardware circuit I have done some research and found out the koblitz mapping method: I will be using a field of binary ...
2
votes
1answer
108 views

What is the difference between order of base point and curve order in EC? [duplicate]

When I was read about the elliptic curve cryptography I found some definition about domain parameter of elliptic curve like the follow. But I did not understand something $p$: prime number. $a, b$: ...
2
votes
1answer
343 views

Why is 224 bit ecdsa faster than 192 bit ecdsa?

I ran several benchmarks using openssl on 2 different computers and I got a surprising result. for the Nist 192 bit curve the benchmark result is ...
2
votes
3answers
273 views

Can a EC private key be derived from a public key?

I understand that the public key does not expose the private key. That is not the question. The question is: Given a EC public key, can a different, but plausible and functional private key be ...
2
votes
1answer
146 views

Is either brainpoolP320r1 or brainpoolP320t1 a SafeCurve?

Not all elliptic curves are safe to use for cryptography, especially from an ECC safety perspective. The site http://safecurves.cr.yp.to/index.html shows that two tested Brainpool curves, ...
2
votes
1answer
211 views

ECC partially blind signature scheme verification

Continued from Is there a flaw in this ECC blind signature scheme? The problem I needed a partially blind signature scheme for one of my projects, but couldn't find one on the internet, so I've made ...
2
votes
1answer
152 views

Verifying multiplicative inverse on a prime field in NIST's ECDSA_Prime.pdf

I am trying to learn about the Elliptic Curve Digital Signature Algorithm (ECDSA) by verifying the results in some example calculations. I found a PDF of example ECDSA calculations from NIST here: ...
2
votes
3answers
227 views

Generating bilinear pairing parameters - running time of finding member of p-torsion group

Update: Question completely rephrased. I want to create the parameters for a bilinear pairing (the Tate pairing in this case). In case you're interested I'm following this thesis, specifically the ...
2
votes
1answer
783 views

ECC - Point Addition/Point Multiplication

So I have a very beginner-esque knowledge of ECDSA and I'm trying to write something in python to take a private key and output the public key (Basically from what I understand just trying to do the ...
2
votes
1answer
240 views

Brute force attack expected running time

I am a bit confused about the expected running times of brute force attacks on different cryptosystems. So let's assume a key size of $2^n$ bits. Symmetric key cryptography: $E(brute)$ = ...
2
votes
2answers
342 views

Scalar Multiplication on Elliptic Curves

In the elliptic curve: $y^2 = x^3 + 20x + 13 \bmod{2111}$. Using the point $P=(3, 10)$ I am wondering how to multiply this point by the scalar $57$? I realize I can write $57*P$ as $2^5*P + 2^4*P + ...
2
votes
1answer
300 views

How are Elliptic Curve Cryptography and Pairing Based Cryptography related?

I have been doing a project that uses the PBC library developed by Ben Lynn. But I am still not clear on how PBC is related to ECC. I know that this is a site for complex crypto QA, but I did not know ...
2
votes
1answer
326 views

How to properly add ECDSA private keys?

I'm currently working on an application that requires me to add two ECDSA private keys in order to make a new private key. The result has to have the property, that its corresponding public key is the ...
2
votes
1answer
64 views

Inversion Free Direct Conversion between Twisted Edwards (X,Y,Z) and Montgomery (X,Z)

The Wikipedia page for Montgomery curves shows how to convert points on a twisted Edwards curve to and from points on an equivalent Montgomery curve. However, their description and the original ...
2
votes
1answer
171 views

ECC vs RSA: how to compare key sizes?

I know and I have understood the details of RSA, elliptic curve cryptography, (EC)DH and (EC)DSA. I keep reading everywhere that (if we don't consider non-deterministic computers) "ECC can achieve ...
2
votes
2answers
145 views

Why are some $x$ coordinates unsuitable for an ECDSA generator point?

For Bitcoin's ECDSA curve (secp256k1, where $a=0$, $b=7$), why can't the generator point's first coordinate be $x=0$? That is, the point on the curve would be $(0,y)$ where $y$ satisfies $y^2 = 0^3 + ...
2
votes
1answer
100 views

Roots in modulo field

I have a point $(X,Y)$ on an elliptical curve $E(a,b)$ where $a=-3$ and $B$ is a large number that is in hexadecimal from -51BD. To compress this point oficially in a program, we know that every $X$ ...
2
votes
1answer
100 views

How many of primitive point on the elliptic curve?

In elliptic curves for cryptography, I know $nG=O$, where $G$ is a base point represented by $G=(x_g,y_g)\ on\ E(F_P)$, where $n$ is Order of point $G$. For example, $P(0,6)$ is a primitive point on ...
2
votes
2answers
114 views

What is primary security in ECC?

I read the following paragraph in a book about Elliptic Curve Cryptography, but didn't understand it: The primary security in ECC is the parameter $n$; where $n$ is Order of point $G$, that is $n$ ...
2
votes
1answer
573 views

Montgomery Ladder vs Double-and-Add

I would like to know what (if any) are the advantages of using Montgomery Power ladder over the Double-and-Add-Always algorithm. I think that firstly, Monty would be slightly faster than DoubleAndAdd. ...
2
votes
1answer
248 views

How is the curve equation used in ECC?

I have a hard time learning exactly how the elliptic curve equation is used in the ECC. $$y^2 = x^3+ax+b$$ If someone knows and could explain to me in simple steps how this is done or a link to it ...
2
votes
1answer
97 views

How do I know if a given curve requires a FpCurve F2mCurve or ECCCurve?

I'm trying to read a public key into Bouncy Castle (secp256k1) and need to choose from the following objects FpPoint; FpCurve; or ...
2
votes
1answer
366 views

Is using Ed25519 parameters in ECDSA safe?

I recently discovered the Curve25519 key exchange lib and the Ed25519 signature lib. Due to the speculations about NIST-designed curves, there is a chance that I ditch them and use the curves above ...
2
votes
1answer
260 views

Elliptic curve cryptography attack vector

I would expect a complicated answer for what seems like a simple question about Elliptic curve cryptography. I've read several entries here such as "Elliptic curve cryptography related key attacks" ...
2
votes
1answer
140 views

Verify Messages to Embedded Device

I'm building an embedded device that I plan on distributing. Periodically the device will poll my server to check for updates and commands. I'd like the device to verify that any messages(JSON strings ...
2
votes
1answer
473 views

How to represent point-at-infinity in affine coordinate

In projective coordinates point-at-infinity can be identified with z=0. How to identify the point-at-infinity in affine coordinate. Whether x=0 and y=0 can be considered as point-at-infinity in ...
2
votes
1answer
1k views

BouncyCastle Elliptic Curve implementation

I'm implementing ECDH key exchange in C# using the BouncyCastle library and I'm having a hard time understanding the elliptic curve side (FpCurve). ...
2
votes
2answers
783 views

ECC algorithm pollard's $\rho$ complexity

One of the methods to break a ECDLP is Pollard's rho algorithm. When ECDLP is defined over a finite field $F_p$, and given a relation $S=w.T$, where S and T are a member of $F_p$. Then ECDLP is to ...
2
votes
1answer
89 views

Double-and-add/Montgomery VS blinding

I'm having a hard time understanding why people use constant-time techniques to counter time-attacks, when blinding seems as good and cheaper to implement. Why do people avoid blinding in ECC?
2
votes
1answer
217 views

What are these twist attacks with cost $2^{58.4}$ on NIST P-224 curve, and when do they apply?

This page on Twist security mentions a combined attack and a twist rho attack, applicable in particular to NIST P-224 curve with cost $2^{58.4}$ something, with no mention precise definition of ...
2
votes
1answer
61 views

Is jacobian to projective conversion unique?

I am doing a small project in ECC. I have used the following equation for converting Projective to Jacobian coordinates: $$D = AC\\ E = BC^{2}\\ F = C$$ and also the following equation to convert ...
2
votes
1answer
57 views

ECDSA signature verifiable 1-way transformations

Alice signs a message $m$ with her private key, yielding a signature ($r$,$s$). I want to prove to someone else that I have this signature, but I don't want them to have the knowledge of what ...
2
votes
1answer
99 views

Implementing AugPAKE over ECC

The AugPAKE spec says it can be implemented over elliptic curves. This sounds very promising, but they don't actually back that claim. Can this really be achieved? If so, how would one go about ...
2
votes
1answer
722 views

Mapping of message onto elliptic curve and reverse it

I would like to perform a variant of Elliptic Curve ElGamal in java using the BouncyCastle libraries. I currently face the difficulty of mapping a message $m$ onto the elliptic curve $E_p$. I have so ...
2
votes
1answer
171 views

ElGamal with elliptic curves I

It is very interesting to see @tylo's answer on ElGamal with elliptic curves. Instead of mapping the message to the elliptic curve point it just reduces an elliptic curve point to its $x$ coodrinate. ...
2
votes
1answer
155 views

Elliptic Curve Verifiable Secret Sharing

I'm reading this paper, which on page 3(Section IV.C) presents a Jointly Random Verifiable Secret Sharing Scheme for Elliptic curves. The algorithm makes sense to me save for this part: "Each ...
2
votes
2answers
431 views

Graphically representing points on Elliptic Curve over finite field

I have taken elliptic curve $E\colon y^2=x^3-4x+20$, defined over $\mathbb{F}_{29}$. The number of points on the curve, $\left|E(\mathbb{F}_{29})\right|=37$. I took base point $P=(1,5)$, and got ...
2
votes
1answer
193 views

Discrete logs on elliptic curve with embedding degree 3 with the 'MOV' attack

The curve $E(\mathbb{F}_{47}):y^2=x^3+x+38$ has order $61$ and $61|47^3-1$ so the embedding degree of $E$ is $3$ and therefore the MOV attack, presumably using some sort of distortion map and a ...
2
votes
1answer
288 views

Choosing good parameter for Lenstra's elliptic curve factorization

In Wikipedia, there is an article explaining Lenstra's factorization algorithm. As far as I got it, we choose some $e \in \mathbb{N}$ and a point $P$ on the curve and then calculate $eP$. While ...
2
votes
1answer
46 views

Second generator for secp256k1 curve

I want to get a group element $h$ on the elliptic curve secp256k1. The important thing is that no one should know the discrete logarithm of $h$ with respect to $g$. That is, $h$ should be created from ...
2
votes
1answer
91 views

Parameters for elliptic curve prime192v3

I'm looking all over the internet for prime192v3's parameters. I think I may have found them here, but it doesn't say what variable each number matches to. Is there some central place where I can find ...
2
votes
1answer
178 views

ECC - ElGamal with Montgomery or Edwards type curves (curve25519, ed25519) - possible?

I know the usual way of using getting shared secrets for encryption with ECC is DH, however, this only works with two keypairs of exactly the same kind, for example two curve25519- or two ...
2
votes
1answer
140 views

Finding Elliptical curve points and encoding text using them

I recently got into learning Elliptical curve cryptography and are currently building a project in C#. Everything is working well so far, I can encode and decode points, and thanks to this forum I ...