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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curve25519 weak points for contributory behaviour

The Diffie-Hellman on curve25519 is usually calculated using the base point $(9,…)$ which induces a cyclic subgroup of $G:=\{\infty\}\cup(E(F_{p^2})\cap(F_p\times F_p))$ with index 8, i.e. there is a ...
2
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172 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 ...
2
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100 views

Why are there only positive value points on an elliptic curve?

I read about elliptic curve cryptography $E$ over $Z_p$ where $p$ is prime number and $G$ is a base point on the curve. I noticed the points resulting from multiplication e.g. $2G$,$3G$,.....,$(N-1)G$ ...
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99 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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114 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 ...
3
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137 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 ...
3
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1answer
112 views

Measure ECC key size

I have implemented a ECC key generation scheme successfully. Now I need to find ECC key sizes of each generating key pairs. I assumed that ECC key size is the size of the ECC private Key. So I would ...
2
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242 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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69 views

Bilinear pairing

I am working on Efficient Construction of Pairings which are being realized by Miller's algorithm. In this algorithm the basic steps are point doubling and line function computation point addition ...
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3answers
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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91 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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120 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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56 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 ...
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1answer
74 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?
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163 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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132 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 ...
3
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1answer
420 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 ...
2
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1answer
140 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 ...
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119 views

Is there a flaw in this ECC blind signature scheme?

Recently I've found the following work on the internet: An ECC-Based Blind Signature Scheme The paper claims to be an ECDSA blind signature however it seems that their scheme has a flaw in it. The ...
3
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1answer
147 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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45 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
243 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 ...
3
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1answer
182 views

Is pairing based cryptography ready for productive use?

I'm currently testing one among those many interesting cryptographic protocols based on bilinear maps. It's quite hard to understand the underlying fundamentals, especially since there are several ...
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1answer
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? ...
2
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1answer
129 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
154 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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235 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 ...
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184 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 ...
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55 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: ...
3
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139 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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1answer
218 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
46 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 ...
2
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70 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 ...
5
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123 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 ...
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118 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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2answers
167 views

Asymmetric key derivation – Who derives the new pub key can't know the new private key

I'm looking for a peer reviewed protocol or algorithm that can do this, preferentially I need something based on elliptic-curves cryptography. Alice know the main public key of Bob (B-PUB-1). Alice ...
5
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2answers
138 views

Is it safe to reuse ECDH asymmetric keys for authentication?

Alice, Bob, and Carol each generate ECDH keypairs. Alice and Bob establish a communication channel and negotiate an AliceBob secret. The question is: Is it safe for Alice and/or Bob to reuse their ...
5
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109 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[] ...
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137 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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256 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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1answer
116 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: ...
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433 views

Elliptic Curves of different forms

Looking at http://safecurves.cr.yp.to/ to find a safe curve, I find that most curves described here are of a different form from that generally used. In Bouncy Caslte, for example, ...
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366 views

Which eliptic curves in OpenSSL 1.0.1f meet all / most of the SafeCurves requirements? [closed]

I am using nginx compiled with OpenSSL 1.0.1f (most current release available). Nginx allows administrators to set a configuration parameter called ssl_ecdh_curve, ...
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1answer
140 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 ...
2
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1answer
333 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 ...
3
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1answer
137 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 ...
5
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111 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: ...
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73 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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129 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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90 views

Weil pairing implementation - low level programming language

I'm just started to studying elliptic curve cryptosystems. My one month-goal is to write a simple signature system based on the Weil-pairing. Some parts of it can be written without deeper ...