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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counting points on elliptic curve

Given an elliptic curve with equation $y^2=x^3+ax+b$, and i want to find the number of points $(a,b)\in E(\mathbb{F}_p)$ where the polynomial has repeated roots, how do i do it? I have an intuition it ...
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126 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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Is there a way to do single key-pair asymmetric encryption?

All of the information on asymmetric encryption I can find uses key-exchanging to get a mutual symmetric key. What I am trying to do is encrypt a message with a public key and have it only readable ...
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349 views

Do Weak Elliptic Curves Exist?

I'm running into very contradictory opinions when try to understand if weak elliptic curves exist. I'm not interested in the case when a curve's weakness is attributed to properties of an EC's prime, ...
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184 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. ...
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101 views

what is the public information in Elliptic curve cryptosystems [closed]

Currently my knowledge about Elliptic curve is quite limited to the textbook and I don't know how a practical Elliptic curve cryptosystem works. I read an example about key exchange using Elliptic ...
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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 $P_2(...
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1k 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. ...
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426 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 ...
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211 views

Impact of algorithms for factoring using elliptic curves over $\mathbb{Q}$

Recently a few papers have appeared that describe a new approach to factoring, using elliptic curves over $\mathbb{Q}$. See, e.g., Factoring integers and computing elliptic curve rational points, ...
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50 views

Favor hash size or field size when systems are disparate?

I'm working on an implementation of Krawczyk's Hashed MQV (HMQV). I'm using Crypto++, which is a C++ library. C++ has some features where classes that represent the crypto objects can be combined ...
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133 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 ...
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235 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 ...
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517 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 ...
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120 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 ...
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350 views

Which is better ECDHE with TLS 1.0

I have a webserver which only supports TLS 1.0 and I am not sure about something: Which is the better cipher in this group when aiming for the best security? ...
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172 views

Finite fields in elliptic curve

I have an elliptic curve defined over finite field where $S_1=aP$ . Is it valid to say that $S_1P$ can also be computed. $P$ is the generator of the group. What my real question is that. Should '$a$' ...
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848 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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108 views

Can a billion elliptic curve keys be generated on a laptop in less than an hour?

I want my application to generate an EC key pair. The first four bytes of the sha256 hash of the public key should contain a known IP address. As hashes are one-way functions, I need to brute force ...
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188 views

Supersingular Isogeny Key Exchange broken?

Found this report detailing a quantum algorithm for computing isogenies between supersingular elliptic curves. http://cacr.uwaterloo.ca/techreports/2014/cacr2014-24.pdf with the quote "...
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175 views

Base point in Ed25519?

The paper "High-speed high-security signatures" by Bernstein et al. introduces the Edwards curve Ed25519. Concerning the base point $B$, it says that $B$ is the unique point $(x, 4/5)\in E$ for ...
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138 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 ...
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463 views

What differences between Menezes–Vanstone ECC and ElGamal ECC?

After researching ECC encryption, I found that we can use ElGamal cryptosystem with elliptic curve and can we use Menezes-Vanstone cryptosystem with elliptic curve. What is the essential difference ...
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146 views

What are differences between $E(F_p)$ and $E(Z_p)$?

When I read some books about elliptic curve cryptography noticed that. sometimes symbolized elliptic curve over $F_p$ is $E(F_p)$ and sometimes symbolized elliptic curve over $Z_p$ is $E(Z_p)$. I ...
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468 views

Security benefits of Ed25519 generating signatures deterministically

I am reading on the Ed25519 curve, and I am trying to understand a claim. Here is the claim: Foolproof session keys. Signatures are generated deterministically; key generation consumes new ...
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ECC public key encryption and authentication - ECIES with ECDSA vs ECDH with AES

I'm currently working on a project where I want to establish a secure and authenticated communication channel between to entities, using Elliptic Curve Cryptography. Now I'm not really sure how to ...
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2k 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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366 views

How to derive formulas for addition and multiplication in Jacobian coordinates

Is there a way to derive the formulas for point addition and multiplication on elliptic curves in Jacobian format by yourself? How could I have derived these formulas by myself?
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842 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 ...
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78 views

tripartite diffie hellman with Weil pairing

I try to understand how the tripartite Diffie-Hellman key exchange works. I read Joux's paper for this: https://www.semanticscholar.org/paper/A-One-Round-Protocol-for-Tripartite-Diffie-Hellman-Joux/...
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73 views

ECFP harder than ECDLP ?

Given two points $P$ and $Q = \sum_{i=1}^{n} x_i.P$ over $E_p(a, b)$ for $x_1,x_2,...,x_n \in \mathbb F_p$. The Elliptic Curve Factorization Problem (ECFP) is to find the points $x_1.P,x_2.P,...,x_n.P$...
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Why inversion and multiplication operations are costly in elliptic curves?

There are several algorithms for efficient scalar multiplication of an arbitrary point P(x,y) by some positive integer k in elliptic curves defined over $F_{p}$ or $F_{2^{m}}$. The scalar ...
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57 views

Are private and public key sizes of Elliptic curve related?

I'm new to elliptic curve cryptography. I just want to know in the case where I take a random number (private key) and find its associated public key, does the size of the public key depends upon ...
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113 views

A (current or soon-to-be) NIST-recommended alternative to ECC?

So this comes from the professional rumor-mill, and I'm wondering if anyone might either debunk or shed light on this. My understanding is ECC is generally now preferred over RSA simply due to how ...
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73 views

How to compromise the privacy of NaCl encrypted messages, when nonce is reused?

I want to know to how break NaCl encryption when a nonce is used more than once for a given key private key. According to PyNaCl docs: It is VITALLY important that the nonce is a nonce, i.e. it is ...
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215 views

Key exchange using ECDH vs ECIES

I'm a beginner to ECC crypto programming. Does any one explain to me the difference with using ECDH for shared key exchange and use of ECIES by encrypting shared key with the public key of the ...
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146 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?
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332 views

Is only one shared secret generated by ECDH per key pair?

I'm confused about ECDH. Using their public keys and private keys, two entities can arrive on a shared secret. But from the equations I've seen, it looks like ONLY the numbers present in their key ...
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184 views

Proving Non-Existence of ECC Backdoors

In light of the NIST Dual EC DRBG scandal, I was intrigued by a NIST slide (slide 9) that said the two points P and Q can be chosen so that the chooser can prove they don't have a backdoor. This ...
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154 views

Twisted curves in protocol

I've come to understand that twisted curves, as for instance defined in the Brainpool specifications, are $F(p)$-isomorphic to their regular $F(p)$ equivalents. So brainpoolP256r1 is isomorphic to ...
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207 views

How to handle the GCD(V,P) != 1 case when doing point addition or point doubling in elliptic curve cryptography

The equation for a finite field Elliptic Curve is of this form: $$y^2 \equiv x^3 + a * x + b \pmod{P}$$ When we do common EC operations like point doubling or point addition we need to calculate the ...
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189 views

Sextic twist optimization of BN pairing - cubic root extraction required?

I found the following paper really interesting: http://www.researchgate.net/publication/220378229_A_family_of_implementation-friendly_BN_elliptic_curves/file/79e4150b3a773beecd.pdf It allows ...
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713 views

Elliptic Curve Encryption Ciphertext Size

I'd like to know how much bigger is the ciphertext when encrypting a message using ECC encrytpion? ECIES (or ElGamal)
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246 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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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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530 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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53 views

Is a hash function used to expand the key after ECC shared secret is complete?

I am trying not to solicit opinions, but I have not been able to find a reference to the question at hand. I have designed an ECC engine in silicon that handles any curve in the form of $y^2 = [ax^3 +...
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130 views

Quantum vs. regular computing time to break ECC?

How long exactly would it take for a regular computer to crack an elliptic curve public/private key via bruteforce, vs. a quantum computer using Shor's algorithm with a couple thousand qubits? Can ...
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41 views

Reuse of TLS client key/certificate in challenge-response protocol

The situation: We have a custom PKI with clients communicating with the server over standard SSL/TLS encrypted channel. PKI uses ECC, server certificate supports ECDHE_ECDSA key exchange mechanism and ...
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77 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 ...