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 many characters per block in an El Gamal ECC cryptosystem?

Looking for the number of characters that can be encrypted using the The elliptic curve ElGamal cryptosystem of each block, I found these lines. But I cannot understand them: Actually in our case ...
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83 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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44 views

Supersingular Isogeny Key Exchange Software [closed]

Are there any optimised implementations of the Supersingular Isogeny Key Exchange by Defeo, Jao, and Plut. It seems like this would be a great drop-in replacement for Diffie-Hellman in both OpenSSL ...
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4k views

ElGamal with elliptic curves

I've searched some information on ECC, but so far I have only found Diffie-Hellman key-exchange implementations using ECC, but I don't want to exchange keys, I want to encrypt & decrypt data like ...
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168 views

Is it possible to choose which point will have the public key of a given Elliptic Curve?

I am wondering if there is a feasible way that, given a specific elliptic curve (such as secp256r1), I could create a keypair where the public key has a given $x$-coordinate. If it is not possible, is ...
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61 views

Is the inverse of a point on an elliptic curve over $\mathbb{Z}_p$ always in the group?

I'm working on a zero knowledge proof system that uses ECC over $\mathbb{Z}_p$ (currently using NIST P-256 since mbed TLS doesn't support group operations on Curve25519, but the problem should be ...
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50 views

Converting ECC Code from python to Java. Extended Euclidean Algorithm not working. [closed]

I'm in the process of converting a Python program I found for calculating ECDSA public keypairs from a given private key. In this particular case it's on the Bitcoin curve. ...
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50 views

PQ Key Exchange based on Elliptic Curve Isogenies

I was reading the Wikipedia article on Post Quantum Cryptography and was interested in opinions as to whether the Supersingular Elliptic Curve Isogeny Diffie-Hellman listed there would be a good Post ...
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202 views

Why does anyone use elliptic curves for a CSPRNG?

I saw Martijn Grooten's talk on elliptic curves at BSides London this year, and it helped me understand how elliptic curve crypto works, especially in the case of Diffie-Hellman (ECDH). He also ...
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151 views

Trying to understand the use of ECC in TLS certificates

I was looking the certificate of this website: https://www.cloudflare.com that has an ECC based certification. I'm just curious to know if is possible to understand which elliptic curve is used and ...
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50 views

Is it right about an example of bilinear map $e$?

The equation for the elliptic curve is $y^{2} = x^{3} + x$ and is defined over the field $F_{q}$ for some prime $q\equiv 3 \pmod 4$, and set $q=307$. Choose random generator $g=[182, 240]$. My ...
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Which attacks are possible against raw/textbook elliptic curve?

A quick question, we know that raw RSA is a no go. To solve this we have different PKCS standards forcing structure on the input messages. For EC the story is something else. For signatures we have ...
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85 views

Would key stretching help mitigate concerns with “verifiably random”?

Daniel J. Bernstein (and others) have expressed concern over how "verifiably random" curve parameters are generated. He points out that hashing a public seed doesn't prevent, say, the US government ...
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2k views

Using ECDSA keys for encryption

I know that ECDSA is used for signature only, but I wonder if I can use the public/private Elliptic Curve keys for encryption too. I have ECDSA SSH public keys and I wonder if I can use them to ...
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42 views

Elliptic curve point addition

I have this curve in E(F131) : $$Y^2 = X^3 + X + 2 $$ I want calculate the sum P + Q considering that $$P= (5,1)$$ and $$ Q = (60, 49)$$ For calculate the result i use these formulas: $$ xr = ...
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102 views

Elliptic Curve Cryptography Encryption and text representation implementation

I'm writing a coursework and right now I've implemented the ECDSA algorithm, but I also need to encrypt and decrypt small text files (.txt) using elliptic curve cryptography. The problem is that I do ...
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162 views

Logjam on Elliptic Curves?

I think we're all aware of the Logjam attack. From now on we know that re-using primes for DH is a bad idea. But we also say that elliptic curves are safe from the attack (relying on the NFS), ...
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Authenticated EC key exchange without a signing/signature scheme?

From my little understanding of EC-based authenticated key exchange protocols, I believe that it is not possible for authenticated key exchange without a signing/signature scheme. Is this correct? ...
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191 views

Given $g^a, g^b, g^c, g^{1/b}$, is it hard to distinguish $e(g, g)^{abc}$ from a random value?

where $g$ is a group element in bilinear group $\mathbb{G}$. I understand it is very similar to the conventional DBDH problem, but $g^{1/b}$ is also known, possibly making it easier? Does anyone know ...
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Severity of Cooking NIST P Curve Constants

Bruce Schneier and Gregory Maxwell have both stated that they believe the constants chosen for NIST's P curves (i.e. P-256r) are cooked. DJB has put together a detailed list of red flags but, outside ...
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205 views

Elliptic curve brute forcing

I have elliptic curve of equation $y^2 \equiv x^3 -x $. And the coordinate of points $Q$ and $P$. I want to solve $Q=[k]P$ (where $k$ is the unknown) by testing all possible $k$. Is this the right ...
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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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47 views

Second generator for secp256k1 curve

I want to get a group element $h$ on the elliptic curve secp256k1. The important thing is that no one should know the discrete logarithm of $h$ with respect to $g$. That is, $h$ should be created from ...
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Why is the P-521 elliptic curve not in Suite B if AES-256 is?

In the NSA's document, "The Case for Elliptic Curve Cryptography", we have ...
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101 views

What does signed fixed window method mean in ECC?

I am studying (sliding) window method in Elliptic Curve Cryptography (ECC) but I am confused by the term, signed fixed window method. By the way term is used in a research paper and not in the book ...
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Non adjacent form of an integer is unique

I have tried to look up the proof for NAF (Non-adjacent form) being unique for every integer, but as far as I have seen, textbooks only mention it as a property of NAF, but no proof is given. Also I ...
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Generating interactive, secure multiple ECC key pairs deterministically

In Elliptic Curve Cryptography (ECC) assuming user A has a private public key pair of $S$, $P$ accordingly with generator point $G$ which as we know is: $$P=S*G$$ Assuming that user A wants to ...
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194 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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635 views

How does ECDH arrive on a shared secret?

I read a brilliant, three part article on Elliptic Curve cryptography (one, two, three). It was able to explain Elliptic Curves to me in a way that didn't require a math degree to understand. The ...
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66 views

Is there any patent free EC point compression available? If not, should I GPL/LGPL my code?

I'm currently using the BouncyCastle implementation in uncompressed mode, and considering releasing an optimized version of EC code that uses point compression. It appears the only thing limiting me ...
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102 views

Is it ever unsafe to compress an EC point?

I am working with a library that outputs EC points in uncompressed form. To save space, I'm considering modifying said library to use compressed EC points. Assuming that I keep track of the sign bit ...
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614 views

Compressing EC private keys

For reasonable security, EC private keys are typically 256-bits. Shorter EC private keys are not sufficiently secure. However, shorter symmetric keys (128-bits, for example) are comparably secure. I ...
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EC based password authenticated key exchange protocol

I would like to know: What is the currently best (efficiency and security wise) known EC based password authenticated key exchange protocol? (i.e. is there any EC based protocol may be similar to DH ...
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69 views

Popularity of ECMQV authenticated key agreement protocol

I've got a few ECMQV related questions: Why is ECMQV a more efficient EC-based authenticated key agreement protocol? Why is ECMQV so popular? Is it because no other protocol of its strength exists? ...
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237 views

Elliptic Curve Crypto, is a distributed signing method possible using Shamir's Secret Sharing?

Note: A distributed signature scheme exists for RSA: Practical Threshold Signatures, Victor Shoup. Is it possible to adapt such scheme for ECC? A centralized signing machine is vulnerable to ...
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Average/approximate difference in value between valid consecutive $x$ coordinates in ECC?

From my basic understanding not all values of $x$ coordinates can satisfy a given elliptic curve equation, i.e. some $x$ coordinate values are not valid points on the curve because $x^3+ax+b$ is not a ...
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71 views

Is there any pattern in points on EC?

I read some where in crypto.stackexchange answer (related to EC-SRP protocol) that there is pattern in points on elliptic curves, i.e. if given some points containing mix of correct and wrong points ...
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132 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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92 views

What aspect of elliptic curve encryption paradigms makes them especially susceptible to quantum based attack algorithms?

This was a statement made during a talk at today's DFN-CERT conference but unfortunately it wasn't explained further. Can anyone shed light on why elliptic curves are susceptible to quantum based ...
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169 views

How does ECDHE_RSA key exchange mechanism work?

Using Wireshark, I found these data exchanged with google.com over TLS: Client Hello possible cipher suites and possible curve types (eg. secp256r1) sent Server Hello cipher suite selected ...
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How can I use SSL/TLS with Perfect Forward Secrecy?

I'm new to the field of cryptography, but I want to make the web a better web by setting up the sites that I host with Perfect Forward Secrecy. I have a list of questions regarding the setup of ...
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How can ECDSA signatures be shortened (to be used as a product key)?

So I made my own serial key generation software, using ECDSA, for use in my own applications and it works great so far! To keep the serial key short enough I use a 128 bit EC curve. My final signature ...
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79 views

What does variable-base point/scalar multiplication mean in ECC?

I am a bit confuse about the term, variable-base point/scalar multiplication, in Elliptic Curve Cryptography. What I have understood so far. It means that the base or point on EC is variable/unknown. ...
2
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1answer
91 views

Parameters for elliptic curve prime192v3

I'm looking all over the internet for prime192v3's parameters. I think I may have found them here, but it doesn't say what variable each number matches to. Is there some central place where I can find ...
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395 views

Safe elliptic curve point addition using projective coordinates: How do I tell if the points are the same?

I am trying to implement elliptic curve point addition in hardware for NIST p256 and p384 curves. I have noticed the following issue with the suggested NIST routines: Consider routine 2.2.7 of ...
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108 views

Koblitz encoding a message to a point, what is the “associated auxiliary base parameter”?

I am looking at the Koblitz method for encoding a message as an elliptic curve point. The first step given in the paper I'm reading is: "Choose an elliptic curve and its associated auxiliary base ...
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260 views

Creating serial key generator using ECDSA, how to get signature short enough?

I've written some questions in this stackoverflow and got great responses but now I'm trying to wrap it all together. I have for the last couple of weeks been building a serial key generator project ...
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52 views

Are there any asymmetric composite order group bilinear pairings?

Are there any asymmetric composite order group bilinear pairings? Is there a drawback of asymmetric over symmetric bilinear pairings of composite order either in efficiency or in security ?
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49 views

In a additive group is it hard to calculate $bg$ given $ag, g, abg$

The ECDH problem defined that given $g,ag,bg$ it is difficult to calculate $abg$. But it is also difficult to calculate $bg$ given $ag,g,abg$. where $g$ is generator and a,b are elements of group.
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90 views

What are the computational benefits of primes close to the power of 2?

Recently I was reading some article about the Bernstein's Curve25519. This is a particular Montgomery curve over $\mathbb{F}_q$ where $q = {2^{255}-19}$. What I missed or was unable to understand is ...