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 ...

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62 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 ...
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58 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 ...
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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 ...
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740 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 ...
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173 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. ...
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161 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 ...
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452 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 ...
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365 views

openSSL ECDH private key size

When you are using a named curve like P-256 in openSSL, is there any standard key size for ECDH private key keys? If you look at the ec_key.c file in the openSSL ...
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196 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 ...
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295 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 ...
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47 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 ...
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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 ...
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188 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 ...
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142 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 ...
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1answer
127 views

Prevent MITM attack while encrypting data by using ElGamal ECC?

I am using ElGamal ECC to encrypt my plain text data. I want to ensure that my data is safe from a Man-In-The-Middle attack. What methodology I can adopt to achieve this goal? How can we prevent a ...
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2answers
212 views

ECDH anonymous key exchange to avoid PKI

I want to use TLS to encrypt the communication between peers in a P2P network. Each peer has a well known 256bit peer identifier (the public key of a 256bit elliptic curve keypair). Both peers need ...
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1answer
305 views

ECC Complexity order of point addition, scalar point multiplication and selecting random point

I am facing this problem in calculating the order of a process which involves ECC point addition: $P+Q$ , scalar multiplication: $aP$, and selecting random points in the group. The group is of prime ...
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371 views

How do the following new (2013) ECC curves compare in security or efficiency? [closed]

I read about the following "safe" ECC curves and notably, secp256 and all the NIST curves are marked as "unsafe" when compared to more modern curves. I need a curve for signing or encryption, (or ...
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2answers
452 views

How do I convert the definition of E-521 into a curve definition a la Bouncy Castle?

I am currently trying to create an ECCCurve for E-521. Unfortunately, it is not currently a named curve in the library I am using, so I will have to define it manually. I am using the definition of ...
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1answer
273 views

Fast modular reduction

I am looking at ways to speed up modular reduction for the polynomial $$2^{256}-2^{32}-2^9-2^8-2^7-2^6-2^4-1$$ I have read the paper "Generalized Mersenne numbers" by J.A. Solinas, but it does not ...
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1answer
100 views

Where can I double check my elliptic curve results?

I am trying to do some elliptic curve calculations by hand, just to refresh myself on how the system works. I calculated some points and did some operations by hand. I am trying to double check my ...
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1answer
202 views

Is this ECC encryption key sharing method okay?

Is this encryption key sharing okay to use? Or is much better to use ECIES? $G$ = base point $a$ = Alice’s private key $b$ = Bob’s private key $A = aG$ = Alice’s public key $B = bG$ = Bob’s public ...
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1answer
315 views

inverse problem about scalar multiplication on elliptic curve

Let $E$ be an elliptic curve over a finite field $F_p$. Given $n$ be a positive integer and $Q$ be a point on $E$, assume that $Q=nP$, how can we find this $P$? We can assume that $n|p-1$. If $n$ is ...
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200 views

EC equivalent for RSA-OAEP

I have some questions regarding aforementioned subject: Is there a EC equivalent of RSA-OAEP key transport/encryption algorithm ? Is ECIES-KEM sufficient ?
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151 views

Discrete log analog of ECM factoring algorithm?

Anecdotally, most factoring algorithms have a corresponding variant algorithm that can be used to attack the discrete log problem using similar ideas. Is there an analog of the elliptic curve (ECM) ...
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180 views

Generating non-supersingular elliptic curves for symmetric pairings

I am looking into the application of pairings in CPABE in particular. I've notice that the scheme uses a supersingular curve as the basis of the pairing. Looking through Ben Lynn's thesis for the ...
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315 views

Are there reference implementations of ECQV implicit certificates?

I am interested in exploring ECC implicit certificates, specifically using the ECQV protocol. While the actual implementation would not difficult to perform using building blocks provided by most ECC ...
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1answer
54 views

Parameter choice Supersingular Isogeny DH

In “Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies” by DeFeo, Jao and Plut (PDF), the public parameters are defined as: Supersingular curve $E$, and bases $P, Q$ ...
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74 views

What happens if no final subtraction is done in Montgomery multiplication?

I'm doing Montgomery arithmetic modulo $N = 2^{255}-19$ for the Curve25519, picking $R = 2^{256}$ for Montgomery. After multiplying two numbers $0 <= A,B < N$ in the Montgomery representation ...
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1answer
81 views

EC: Why does $h>200$ need to hold?

The class number of the principal order belonging to the endomorphism ring of E SHOULD be at least 200. From TR-03111 (pdf page 15). This value commonly is referred to as $h$. So my ...
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60 views

Which attacks are possible against raw/textbook elliptic curve?

A quick question, we know that raw RSA is a no go. To solve this we have different PKCS standards forcing structure on the input messages. For EC the story is something else. For signatures we have ...
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146 views

As a cryptographer, what are the things I should care about in my implementation of pairing functions?

As a beginner in cryptography, I do not know anything about different pairing types more than their names. So far, I know these names: Ate pairing, tate pairing, eta pairing, and r-ate pairing. I am ...
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279 views

Elliptic Curve based blind signature implementation

I want to use Elliptic Curve based blind signature scheme for my research. There is no proper implementation of ECC-based blind signatures. Can someone describe to me which things I need to follow ...
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76 views

Reliability of a single-pass deniable authentication protocol?

I look for one-pass deniable authentication protocol with a short message payload for my project and find a solution: ...
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0answers
175 views

Defending hybrid encryption schemes against padding oracle attacks

I intend to use a generic integrated/hybrid encryption scheme for transmitting information between a client and a server. Key encapsulation: a 128-bit symmetric key is generated and asymmetrically ...
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0answers
120 views

Gallant-Lambert-Vanstone method

I am experimenting with the GLV method but cannot manage to get the right results according to the literature. I managed to find lambda, beta, split $K$ into $k_1$ and $k_2$ etc. for the curve I'm ...
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148 views

Known vulnerabilities in (EC-)KCDSA

Does anybody know if there's known vulnerabilities in KCDSA/EC-KCDSA? I have been researching for the past few hours and I haven't found anything. Wikipedia has very limited amount of information and ...
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2answers
111 views

Severity of Cooking NIST P Curve Constants

Bruce Schneier and Gregory Maxwell have both stated that they believe the constants chosen for NIST's P curves (i.e. P-256r) are cooked. DJB has put together a detailed list of red flags but, outside ...
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3answers
217 views

Which area of Maths should I pursue?

I would like to know which area of Mathematics would be most beneficial to cryptography. Surely Algebraic Number Theory and maybe to a lesser extend, Elliptic Curves, are closely linked to ...
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1answer
130 views

Simplified Example of ECC to use in the classroom

I have come up with the following rudimentary example of how ECC relates to asymmetric keys. Is this a valid explanation of ECC and its relationship to asymmetry? To only be deciphered by the person ...
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193 views

Elliptical curve cryptography key generation time

I am currently trying to learn more about Elliptical curve crypthography and have finally started to get things working and undestanding the different pieces. I've written a small project in C# and ...
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2answers
175 views

64 bit Elliptic Curve key?

For a simple proof of concept project i'm (attempting!) to do, i've started looking into openSSL elliptic curve cryptography. However instead of the standard key lengths, 160-512. I'm interested in ...
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1answer
94 views

Does a cofactor of an elliptic curve have to be an integer?

What are the implications of a curve having a non-integer co-factor for its generator point? Is that even possible?
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183 views

Is there a problem with this non-ECDSA message signing?

Is there a problem with signing a message like this? ...
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2answers
205 views

Elliptic curve brute forcing

I have elliptic curve of equation $y^2 \equiv x^3 -x $. And the coordinate of points $Q$ and $P$. I want to solve $Q=[k]P$ (where $k$ is the unknown) by testing all possible $k$. Is this the right ...
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118 views

Explanation of each of the parameters used in ECC

I'm having a very difficult time finding a clear explanation of the parameters used elliptic curve cryptography. I know for certain that $p$ is the number or order or whatever of the given field that ...
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147 views

Why does NaCL have different keys for signing and encryption?

I want to start using NaCL to sign messages that will go into a message queue, and I noticed that it generates different keys for each operation. Is there a reason for this? Can I not use the same PK ...
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1answer
121 views

Three-way key exchange with elliptic curves without pairing

Assume that there are three users, each with their own secret key $d_i$ and the corresponding public key $Q_i = d_i \cdot P$, such that $Q_i$ is a point on an elliptic curve and $P$ is a base point on ...
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1answer
329 views

Efficient algorithm for remainder calculation over prime field for ECC implementation?

I am working on 224-bit elliptic curve cryptography. In this 224-bit * 224-bit multiplication results 448-bit output. I am reducing 448-bit into prime field range( prime number $2^{224}-2^{96}+1$) ...
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3answers
448 views

File encryption with one keypair?

I'm working on a program that uses an ECC keypair in a (password protected) PKCS12 file (.pfx) to encrypt files. I like this method because I think it will be higher security (using ECDH to negotiate ...