Questions tagged [elliptic-curves]

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 consider more specific tags such as discrete-logarithm and ecdsa.

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8
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2answers
723 views

Rely on NSA Suite B Cryptography?

NSA's Suite B Cryptography suggests some cryptographic algorithms for encryption, digital signatures, message digests and key agreements. The selected algorithms and their key size are suggested by ...
3
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1answer
1k 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 ...
2
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0answers
207 views

Hardware Implementation of Pairing over BN curves

I am in the middle of FPGA based Hardware architecture design for the computation of Pairing (particularly R-ate Pairing) over BN curves. Where, the point addition, and point doubling should be ...
6
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1answer
2k views

Converting Ed25519 public key to a Curve25519 public key

I understand that: $$x_{montgomery} = \frac{1 + y_{edwards}}{1 - y_{edwards}}$$ Using the libsodium ed25519 implementation, I have tried to write the following: ...
5
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1answer
360 views

Implementing CD serial key system

I am trying to create a system where to unlock the application one needs to enter a serial code. I have read many articles on the theme but there are two problems bugging me. One is, If I have a ...
2
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1answer
367 views

Adding points on Elliptic Curves

How do we add the integer points $P=(-1, 4)$ and $Q=(2, 5)$ on the elliptic curve of the form $y^2=x^3+17$ ?
5
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1answer
409 views

Elliptic Curve Cryptography

I have been trying this for a while. But I couldn't get it. How can I determine the point of intersection of the tangent line at (0, 0) on the curve $y^2 + y = x^3 + x^2$ ?
5
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1answer
1k views

How can I find the order of the group that an elliptic curve is defined over?

I have a Weierstrass elliptic curve ($y^2=x^3+a \times x+b \mod p $) How can I find the order of the group itself? I have seen Mathematica has a GroupOrder[] ...
0
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2answers
1k views

Elliptic curve group over a prime finite field $F_p$

If $p$ is a big prime, and the elliptic curve $E$ is defined over $F_p$ by the equation $y^2=x^3+ax+b$ where $a,b\in F_p$. The point on $E/F_p$ together with the infinite point $\mathcal{O}$ form a ...
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2answers
302 views

pairing-based schemes

some authors claimed that computational performance of a pairing-fee scheme (based on scalar multiplication over an elliptic curve group) is about 1000% more efficient than a pairing based one I would ...
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0answers
263 views

A question about elliptic curves and finite fields in bilinear pairings

Based on what mentioned in the paper "Pairings For Cryptographers" http://www.sciencedirect.com/science/article/pii/S0166218X08000449 the two inputs of a pairing map are two members of two additive ...
2
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1answer
189 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 ...
4
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1answer
779 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, ...
1
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1answer
1k 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 ...
3
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1answer
186 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 ...
5
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1answer
923 views

Parallelized Pollard's Rho algorithm for ECDLP + Jacobian coordinates

My implementation of the parallelized Pollard's Rho algorithm is using Jacobian coordinates to avoid the costly inversion operation when performing point addition. I am wondering if there are any ...
4
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1answer
1k 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 ...
2
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1answer
565 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
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1answer
200 views

Integers in ECC

Let A be a point on curve with integral coordinates. Does k.A necessarily have integer coordinates? If so than why and if not than how to find A and k such that k.A has integral coordinates.
3
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1answer
219 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 ...
1
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3answers
279 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)?
6
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1answer
3k 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. ...
1
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1answer
910 views

Smart Card Basics

I want to implement some of the basic encryption algorithms on smart card, could any body guide me how to program a smart card, which tools (hardware and software) I should have, and if these tools ...
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0answers
88 views

Want to use ECC but am clueless [closed]

First off, I'm not an experienced cryptography or computer person, please bear with me. I have some basic experiences with PGP software though (not much of a redemption huh?). I have some data that I'...
3
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1answer
392 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 ...
1
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1answer
187 views

order of elliptic curve divisible by prime

Why order "u" of an elliptic curve "E" defined over a finite field "GF (q)" (u = | E / GF (q) |) must be divisible by a large prime number r to be appropriate for cryptographic purposes?
2
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1answer
2k 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 ...
0
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1answer
439 views

Is C25519/Ed25519 “twist secure”?

This recent new curve mentions something that's new to me: twist security. http://safecurves.cr.yp.to/bada55.html Are the existing C25519/Ed25519 curves secure against this form of attack?
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1answer
340 views

Is this EdDSA modification secure?

I am hoping to employ a signed set membership system which is valid iff each signer's contribution to the set is present. The system should allow for two or more mutually exclusive signed sets to be ...
9
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1answer
4k views

ECDSA vs RSA: Performance on Android platform and surprising results

For our privacy-preserving protocol, an encrypted channel is established. In order to protect our system from man-in-the-middle attacks, signature-based approach is used. After we've implemented it ...
3
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1answer
836 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 ...
4
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1answer
1k 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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0answers
70 views

Smart card Strong authentication / Verification ( fingerprints)

I'm trying to make a strong authentication software and embedded software in a java card. I have found many papers and publications about the subject… too much information to process and I'm working ...
3
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1answer
2k views

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 ...
5
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1answer
592 views

Are RSA or ECC vulnerable to an attack where the same (unknown) plaintext is encrypted with multiple public keys?

I'm not sure what this attack model is called - it's not known-plaintext and also not quite cipher-text-only. It is similar to this question except the general case (not just two keys) and using keys ...
1
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1answer
100 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? ...
2
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1answer
910 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 \...
6
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2answers
4k views

Are there any elliptic curve asymmetric encryption algorithms?

RSA offers the functionality of encrypting (short messages, or symmetric keys) with a public key, and decrypting with a private key. However, RSA key generation is extremely expensive, especially for ...
7
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1answer
796 views

Is there an algorithm to check if an elliptic curve is secure?

As I understand it elliptic curves are of the form $y^2 = x^3 + ax + b$ Where $a$ and $b$ are the curve parameters. However not all parameters will give a curve suitable for crypto purposes. Is there ...
2
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0answers
106 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: ...
5
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1answer
503 views

Side-channel attacks against ECDH for Weierstrass normal form curves

I hear a lot about why Montgomery curves are used in ECC, and one reason is that the same algorithm can be used to do both point addition and doubling (this is not true for the Weierstrass normal form)...
4
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0answers
493 views

Using the same private key for two ECC key pairs

Let $(d_1,Q_1)$ and $(d_2,Q_2)$ be ECC key pairs over two different elliptic curves (say NIST P-224 and NIST P-256). According to the Elliptic Curve Discrete Logarithm Problem (ECDLP), if the private ...
2
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0answers
376 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 ...
3
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1answer
195 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 ...
7
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2answers
815 views

Besides key and ciphertext sizes what are other advantages of elliptic curve versions of various protocols?

There are elliptic curve variants of Diffie-Hellman, ElGamal, DSA and possibly other protocols/algorithms. I know that these elliptic curve variants have smaller key and ciphertext sizes which will ...
4
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1answer
758 views

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 ...
9
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1answer
3k 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
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1answer
400 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: ...
5
votes
1answer
227 views

cryptographically good random elliptic curves?

After an answer here, about generate elliptic curves, I've start thinking about the algorithm. The mentioned algorithm will produce curves in the Weierstraß Reduced Form (WRF) over finite fields: $...
1
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2answers
130 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 ...