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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How to calculate key size of a security algorithm?

There are lots of security algorithms. One of the way to measure security of a cryptography algorithm is to find out its key size. There are many key size of a single algorithm. ECC (Elliptic ...
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Do most TLS 1.2 implementations express curves in a canonical form when performing EC arithmetic?

Sorry if this is a silly question, but does anyone know if the cryptographic libraries which implement TLS 1.2 for Firefox, Chrome, etc. express a given curve in a canonical form (i.e. one of ...
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41 views

ECDH-ECDSA Combination

I am doing research on cryptography primitives at a basic level and I faced a question on encryption methods. I understood that ECDH is an approach to for secure key exchange between two parties ...
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43 views

construction of division polynomials

I'm not sure if this is the right place for my question.... I'm trying to understand the construction of the division polynomials used in the schoof algorithm. I firstly followed the report of ...
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1answer
31 views

Can the backdoor in Dual_EC_DRBG be used to create a public key stream cipher?

Dual_EC_DRBG has the property that if $Q = e\cdot P$, someone who knows $e$ can break the PRNG. This seems to lead to a public-key stream cipher: Alice chooses a random $P, e$, where $P$ is a ...
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45 views

Elliptic curve ElGamal with homomorphic mapping

I am interested in ElGamal due to the fact that you can achieve some degree of homomorphic properties. I became interested in applying ElGamal to elliptic curves, and found this other question with an ...
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56 views

SHA1 collisions and the impact for ECDSA signatures

What will it mean for ECDSA using SHA1 when we have practical attacks breaking the collision resistance property of SHA1? [UPDATE] Added a bit more details to be clear. If $(r,s)$ is the ECDSA ...
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31 views

Point Multiplication Double-And-Add-Method [migrated]

Double and Add Method If i have k=5, base poin G =(2,1) Is is double and method like this: 5.G = 2.(2G)+G How is the logic, is i want convert integer 5 to base 2, like this : 5 = 101 ...
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23 views

Degrade in performance with SSL_OP_SINGLE_ECDH_USE?

We have used SSL_OP_SINGLE_ECDH_USE when setting up our SSL_CTX . This seems to be causing a degrade in performance. I'm not able to find proper documentation for it except that it generates new ...
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153 views

Difference between Pseudo Mersenne primes and Generalized Mersenne primes

The field prime numbers $p$ proposed by the NIST standards are referred to as Generalized Mersenne prime numbers [1] and as Pseudo Mersenne prime numbers [2]. Is there a difference between Pseudo ...
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25 views

Double add always + scalar blinding [migrated]

I am using double add always + scalar blinding for pointMultiplication in projective coordinates (x,y,z). poinMultiplication Class ...
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112 views

Finding the subgroup in isogeny-based cryptography

Isogeny-based cryptography is one of the newest post-quantum cryptography. Hardness of this system is based on finding isogeny between two elliptic curves. Also this is theorem: Elliptic curves ...
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EC Schnorr or Ed25519 Java card implementation [migrated]

I am trying to implement ED25519 in java card. Since there are many restrictions with respect to data types and sizes, its becoming tough to implement it, especially curve operations. Is there any ...
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39 views

Example of Projective Coordinates

Given the affine form of coordinates $(x,y)$ such as $(5,3)$, if I want to convert $(5,3)$ to projective coordinates $(x,y,z)$, should the form of point be $(5,3,1)$? It is triplet not a point, right? ...
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30 views

Convert projective to affine coordinates in ECC? [closed]

I am working with my project. I use projective coordinates but when I convert to affine coordinates, I can't get it. Can anyone help me? Projective Coordinates $(X,Y,Z)$ to Affine Coordinates ...
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5 views

Different Signature from Affine to Projective Coordinates [migrated]

My project is compare ECDSA performance between Affine and Projective Coordinates. I'm using BouncyCastle API in Java. Part of Main ECDSA class code: ...
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54 views

Semaev summation polynomials

I am little confused how this attack works. We have the points $P, Q$ such that $Q = nP$. We let $u_{1} $and $u_{2}$ such that $R(x,y)=u_{1}P+u_{2}Q$. Then if we find the solution $x_1,...,x_n$ of the ...
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1answer
91 views

Limitations of Elliptic Curve Cryptography?

Simple question, what are the limitations of ECC, both in terms of application and how secure it is? I heard that the NSA were able to read emails a few years back due to a backdoor they had ...
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20 views

Using Montgomery ladder to calculate the coordintes

In one of my assignments I need to solve the below: For a Montgomery curve $3v^2 = u^3+u^2+u$ over ${\mathbb{F}}_{11}$ and point $P = (9,8)$. I need to compute $x$ coordinate of $3P$ using ...
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Are all points in a cyclic group in ECC pre-calculated?

I think I now understand the majority of the ECC protocols, however I'm slightly unclear on some of the finer details? So we need the cyclic group formed by the generator point, but this is a very ...
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68 views

Can we break ECDLP with this machine?

Let $P$ and $Q$ are two points of NIST elliptic curve $E$ (defined over $F_{2^m}$ with prime $m$) and $k$ is a private key such that $k.P=Q$. Also we have a machine that is able to leak some ...
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Performance of ECDSA, ECKCDSA and ECGDSA

It is proven that ECDSA algorithms are faster in key and signature generation compared to RSA. In addition, the signatures are much shorter. However, I would like to know the performance difference ...
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Elliptic ElGamal Public Key Cryptosystem doubt

I need an example of Elliptic ElGamal Public Key Cryptosystem. I have been trying with some values but I don't get the right solution. I have $p=13$, the elliptic curve $E:y^2=x^3+11x+7$ and a point ...
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What is th purpose of m and q in elliptic curve cryptography protocols?

In crypto protocols that contains calculus on elliptic curves I can often see $\dfrac{m}{q}$$Q$ where $m$ stands for order of EC points group and $q$ is the order of corresponding subgroup of $m$. $Q ...
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Convert messages to elliptic curve points [duplicate]

Let $E$ be an elliptic curve; $\alpha,\beta$ two points of $E$; and $a$ a private key such that $\beta=a\cdot\alpha$. We choose random integer $k$ and plain text $x\in E$. Encryption and decryption ...
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Default algorithm for scalar multiplication of elliptic curve points by the MIRACL Library

What is the default algorithm used by the MIRACL-Library [1] for elliptic curve cryptography systems to perform scalar-point multiplication with curves of Weierstrass form satisfying the equation : ...
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86 views

Counting points on elliptic curve over binary field

How to count number of rational points on elliptic curve over binary field?
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2answers
197 views

Why is 2048-bit RSA always paired with 320-bit ECC?

You may already have noticed that most smart cards ship with 2048-bit RSA support and 320-bit ECC over GF(p) support. You may have already asked yourself "why exactly 320-bit?". Now I remember having ...
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2answers
223 views

Curve25519 vs “Million Dollar Curve”

Quoting from the Million Dollar Curve website: By using publicly verifiable randomness produced in February 2016 by many national lotteries from all around the world, we propose to generate a ...
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2answers
383 views

Can elliptic curve (25519) be used to encrypt file?

This is probably a simple question, but I haven't been able to see it stated anywhere. Is it possible to directly encrypt a file (of any length) with some form of EC using the 25519 curve. I know ...
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1answer
89 views

How does encryption work in elliptic curve cryptography?

So I think I understand a good amount of the theory behind elliptic curve cryptography, however I am slightly unclear on how exactly a message in encrypted and then how is it decrypted. So my ...
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1answer
54 views

Schnorr Ring Signatures - wrong hash results

I don't know if I've placed the question right, It is half maths, half programming. I'm writing Schnorr Ring Signatures on Elliptic Curves in Java and I have a problem. I've found scheme on integers ...
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Do I understand (below) why Q = dP is easy while finding d is hard

As we all know for discussion of Dual_EC_DBRG, the point on an elliptic curve Q can be calculated from P and some (large) integer d $Q = dP$ And we know that knowledge of Q and P is not sufficient ...
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What happens if you multiply a point with its order on complete Edwards curves?

I was recently working with some ECC crypto and stumbled across the following phrase on the SafeCurves page: The rational points of a complete Edwards curve are the pairs (x,y) of elements of ...
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4answers
202 views

What is the most secure ECC Curve?

I have for a while used Koblitz curve (sect571k1), in ECDH and ECDSA. But I have started wonder if it is the most secure. I prefer security over efficiency. So the curve doesn't have to be the most ...
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How do I get the equivalent strength of an ECC key?

I know how to calculate the comparable symmetric strength of an RSA modulus: calculate the running time for a field sieve. This is how NIST gives approximate symmetric sizes for asymmetric algos in ...
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48 views

HD (Hierarchical Deterministic) Keys using Safe Curves?

Bitcoin's HD (Hierarchical Deterministic) Keys as described in BIP32 allow for a master key to be created (a private key and a chain code) such that a tree of both public and private keys can be ...
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40 views

Translation of Schorr Ring Signature to ECSchnorr Ring Signature

I have to write an EC version of Schnorr Ring Signature Scheme. I've already wrote regular ECSchnorr Signature Scheme using this (page 128). I've found a scheme of Schnorr Ring Signature Scheme (page ...
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1answer
62 views

Elliptic Curve ElGamal and DSA - smooth group order and element of large prime order

In regular ElGamal and DSA, we choose large primes $p$ and $q$ such that $p\equiv 1\pmod{q}$, and a group element $g$ of order $q$ by computing $a^{(p-1)/q}$ for some random $a$. This is to prevent ...
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Seeking an implementation of the Satoh algorithm for elliptical curve point counting

I would be very grateful if someone has an implementation of the Satoh algorithm (Fast Elliptic Curve Point Counting). Can someone point me to practical algorithm implementations or provide some ...
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0answers
60 views

How to compute projective cordinate Z in elliptic curve cryptography?

I was working on affine coordinates and struggling with the computation time taken for operations and then I was advised to use projective coordinates so that mul-inverse operation can be avoided Can ...
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31 views

Is it possible to re-generate ECPublic key from raw private key and curve params?

As per my understanding, in Elliptical curve cryptography, the EC PrivateKey is nothing but a bigInteger value. If you know the curve specs and you have this private BigInteger value (which I am ...
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3answers
113 views

Is there any alternative for extended euclidean algorithm to perform modulo division?

I'm implementing point addition and point doubling operations for elliptic curve cryptography. I have implemented extended euclidean algorithm to perform modulo division. It appears the that ...
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1answer
65 views

How to perfrom modular division while numerator is lesser than the denominator?

I'm implementing point addition and point doubling of elliptic curve cryptography. The formula that I'm using for slope is Point addition: $S = \frac{(P_y-Q_y)}{(P_x-Q_x)}$ where $P$ and $Q$ are the ...
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1answer
63 views

Optimal same-base exponentiation?

I've (finally) implemented the answer to this question in our library, which stated how to transform montgomery curves (and points) to weierstrass curves (and points). Now, for scalar multiplication, ...
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elliptic curve multiplication with negative factor

I'm learning the multiplication operation on EC. From most material I can found, the multiplication $nP$ is just: $$nP=P+P+\cdots +P+P$$ For negative factor, i.e. $(-n)P$, by above definition and the ...
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A standard extension of ECIES for multiple recipients (broadcast / multiparty)?

I have one sender, and a small number (~5) of recipients. The sender knows each recipient's public EC key. I want the sender to broadcast a single message in such a way that any one of the recipients ...
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Cryptographic operations for NISTP256 can be implemented using montgomery method?

I need to implement cryptographic operations starting with the curve NISTP256. I have been told to implement it using Montgomery method. I read a pfd from ...
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RSA & DH at risk due to math advances, will this eventually affect elliptic curves too?

I was looking into the predictions by some researchers that RSA and Diffie-Hellman may not be secure in the next few years due to advances in math and being able to calculate the discrete logarithm ...
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25 views

Edwards / Montgomery ECC over binary extension fields

I recentely had a discussion about the redesign of our ECC code for the library I'm collaborating on and the person I was discussing with came up with Edwards and Montgomery curves over binary ...