Elliptic curves are a mathematical structure. In cryptography, it is common to use the structure $y^2 = x^3 + ax^2 + b$ over a finite field. Questions relating to elliptic curves and derived algorithms should use this tag and might also consider specific tags such as discrete-logarithm and ecdsa.

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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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3answers
249 views

understanding pairing $e:G \times G \to G_T$ and ( Decision)BDH assumption

From DrLecter's comment, I know that DDH problem can be efficiently solved with this $$e(g^a,g^b)\stackrel{?}{=} e(g,g^z).$$ I have some trouble to understand this map $e:G \times G \to G_T$. Am I ...
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231 views

While generating a random Elliptic curve what are the conditions i have to considerd?

I want to generate a random elliptic curve over a prime field. What are the conditions I should satisfy? For the NIST recommended standard ECC-224 bit curve with prime $p=2^{224}-2^{96}+1$, a ...
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124 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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215 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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1answer
887 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). ...
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1answer
58 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
90 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
124 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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1answer
134 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
90 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
319 views

Adding and multiplication in jacobian coordinates

How can I derive formulas for adding 2 points and multiplication by a scalar in Jacobian coordinates $(x,y) = (\frac{X}{Z^2},\frac{Y}{Z^3})$ over an elliptic curve?
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175 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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1answer
130 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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1answer
154 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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1answer
228 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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67 views

Difference between “ECDH with cofactor key” and “ECDH without cofactor key”?

I need to use “ECDH with a cofactor key” for generating symmetric key. I have a fair idea on how ECDH works, but I don’t understand the cofactor part. What is the difference between ”ECDH with a ...
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158 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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78 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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0answers
117 views

Can keys from Bitcoin's Hierarchical Deterministic Wallets be correlated (reducing privacy)?

I'm trying to understand if the feature "Hierarchical Deterministic Wallets" in Bitcoin allows for complete privacy of all derived keys, and if any of those keys can be associated with each other ...
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317 views

Understanding elliptic curve encryption

I'm having a hard time understanding the elliptic curve encryption. One thing thing I don't understand is listing all the points on the curve (mod p). Suppose I have the following elliptic curve: y^2 ...
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134 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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199 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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3answers
187 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
135 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 ...
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125 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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83 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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2answers
144 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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1answer
100 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, ...
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1answer
112 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
235 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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324 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 ...
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367 views

ECIES protocol - what does the || operation mean?

I am studying elliptic curves problems, which also includes study of related protocols such as ECIES. A there is a problem I don't understand operation $||$. What this operation mean? Some stuff is ...
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1answer
409 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 ...
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2answers
247 views

ECM Implementation is really slow

I followed the algorithms 14.4 (computes 1st and 3rd coordinates in (X,Y,Z)#k modulo n) and 14.5 (factorization using ECM) in David Bressoud's book 'Factorization and Primality Testing'. I think the ...
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100 views

Encryption time in ECC

In RSA, encryption time is usually much less than decryption time due to having a small public exponent. Can this be achieved in Elliptic Curve Crypto (ECC)?
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56 views

Why does knowing the number of points on a curve help solve ECCDLP?

Perhaps, this is a really obvious question, but I am still having trouble understanding how this all fits together. Why is knowing the number of points on an Elliptic Curve helpful in cracking it? ...
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1answer
148 views

ECDSA Public Key generation

Referring to both Wikipedia page and ECDSA-cert paper I can understand that, given $\mathcal{E} = \mathcal{E}(a,\,b,\,\mathbb{F}_{2^m})$ as our elliptic curve on $\mathbb{F}_{2^m}$ group $G \in ...
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70 views

Finding an x such that xP = (11,44) on an elliptic curve

Given the elliptic curve $$E:y^2 = x^3+17x+5 \mod 59$$ with point $P = (4,14)$, how do I find $x$ such that compute $x\cdot P = (11,44)$ Is there a mathematical method to compute $x$, or do I ...
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1answer
99 views

If its possible to derive the public key from a private key, why can't we go in reverse?

I'm looking at source code for BitcoinJ that derives a public key from the private key. ...
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1answer
143 views

What are some restrictions when converting Montgomery Curves into Weierstrass Curves?

I want to represent a Montgomery Curve (curve25519) in Weierstrass form as a personal exercise. After doing some math and referencing the conversion equation at ...
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1answer
373 views

Signature algorithm SHA 1-2 with ECDSA

Can someone please explain what key sizes are required for the ECDSA algorithm? I tried a 128 bit EC Key for SHA1withECDSA and it throws an error. However with 256 bit key I could run the algorithm. ...
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154 views

With TLS and ECDHE, how does curve selection work?

Given that a TLS client and server have already agreed upon ECDHE for session key establishment, how does the selection of the actual elliptic curve ("domain parameters") being used for deriving the ...
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1answer
115 views

Efficiency of computing $e(P,Q)$ Vs $g^a \pmod{p}$?

The definition of $e$ can be seen here. I want to know the accurate comparison of efficiency between $e(\cdot,\cdot)$ and $g^a \pmod{p}$. If computing $e(P,Q)$ is less efficient than computing $g^a ...
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1answer
195 views

Can Secp 256 K1 curves “map” to a value on FIPS 186-3 or P-256?

I'm looking at Secp 256K1 vs UProve's FIPS 186-3 or P-256 implementation. Is there any relationship between the curves such that I can consistently "map" or "project" values from one curve to ...
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1answer
188 views

Relationship between Elliptic Curve Discrete Log, Integer Discrete Log, and Integer Factorization

I am trying to look into a relation between the following three problems which are widely used to build public crypto systems: Integer Discrete log Elliptic Curve Discrete log Integer Factorization ...
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196 views

ECC Point Multiplication of Product

I can calculate $Q = a\,b\,G$ in several ways: $Q = a \, (b \, G)$ or $Q = b \, (a \, G)$. These give the same result, as expected. But if I do $c = (a \, b) \bmod n$ where $a \, b$ is much greater ...
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127 views

What are unified addition and differential addition in elliptic curve point arithmetic?

A lot of papers use these terms but I do not find a proper explanation of them. Can somebody tell the meaning / difference / intuition / application and if possible with an example.
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39 views

ECIES: Purpose of MAC?

Assumptions: $A$ wants to communicate with $B$ $A$ knows a public key $P_B$ which is trusted by a third party and belongs to $B$ $A$ knows the address of someone who pretends to be $B$ $A$ wants ...
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1answer
94 views

How to convert projective to jacobian co-ordinate in ECC?

I am doing a small project using elliptic curve in cryptography. My doubt is, can I directly convert a projective to a Jacobian coordinate system without using the affine conversion in elliptic curve ...