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 ...

learn more… | top users | synonyms (2)

16
votes
1answer
994 views

Mapping points between elliptic curves and the integers

My primary question is: Is there an easy way to create a bijective mapping from points on an elliptic curve E (over a finite field) to the integers (desirably to $\mathbb{Z}^*_q$ where $q$ is the ...
6
votes
1answer
239 views

Difference between “ECDH with cofactor key” and “ECDH without cofactor key”?

I need to use “ECDH with a cofactor key” for generating symmetric key. I have a fair idea on how ECDH works, but I don’t understand the cofactor part. What is the difference between ”ECDH with a ...
3
votes
1answer
190 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 ...
10
votes
1answer
2k views

What are the advantages of a static ECDH key?

What are the advantages of using "static-ephemeral ECDH" over "ephemeral-ephemeral ECDH"?
1
vote
1answer
92 views

Why we need ECDSA when we have ECDH?

ECDSA and ECDH give us the following methods: ...
2
votes
1answer
470 views

openSSL ECDH private key size

When you are using a named curve like P-256 in openSSL, is there any standard key size for ECDH private key keys? If you look at the ec_key.c file in the openSSL ...
2
votes
1answer
116 views

Can I use an ECDH Shared Secret from the same Private / Public Key Pair?

I'd like to know if using the ECDH shared secret of a static EC Private Key with it's own corresponding static EC Public Key causes a problem / weakness. (edit) not asking if it's ok to re-use the ...
2
votes
2answers
327 views

ECDH anonymous key exchange to avoid PKI

I want to use TLS to encrypt the communication between peers in a P2P network. Each peer has a well known 256bit peer identifier (the public key of a 256bit elliptic curve keypair). Both peers need ...
4
votes
1answer
398 views

EC ElGamal versus static+ephemeral ECDH

A client application needs to encrypt a UDP datagram for a server with known EC public key $P$. Performing a full ECDH key exchange would defeat the benefit of using UDP as a connectionless protocol. ...
1
vote
0answers
41 views

Clarify EC point addition and multiplication

Please clarify the below doubt regarding EC point addition and multiplication: $P$-Generator Point; $a$ and $b$ are integers; $X$ and $Y$ are EC points, defined as follows: $X = (a*P) + (b*P)$ $Y = ...
9
votes
1answer
178 views

How many bits of entropy does an elliptic curve key of length n provide?

A FAQ for an open source project makes the claim: Indeed, an elliptic curve key of length n provides $n/2$ bits of security. I have two questions: What is the practical difference between ...
1
vote
0answers
102 views

Is this a secure method of encrypting with authentication?

My goal is to allow two clients to send files securely over an untrusted network without the need for more than one block of information to be sent. Both clients have ECDSA keys of size 256 bits. I'd ...
1
vote
1answer
42 views

Question on Miller's algorithm (change the input m)

From the book titled " An Introduction to Mathematical Cryptography" (Chapter 5,page 322), we know that the miller's algorithm returns a function $f_P$ whose divisor satisfies $$div(f_P) ...
2
votes
1answer
136 views

Ring signatures in ECC

Are ring signatures possible with elliptic curves? If so how. The original paper by Rivest, Shamir, Tauman seems to require an invertible trapdoor function. But I've only seen algorithms for secret ...
5
votes
1answer
197 views

EdDSA Signature Algorithm - hash of secret key

Why does EdDSA use the (SHA512) hash of the secret key as the exponent for the public key rather than using the secret key value directly? This seems inefficient and I can't see how it adds any extra ...
3
votes
1answer
123 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 ...
3
votes
1answer
299 views

ECC - ElGamal with Montgomery or Edwards type curves (curve25519, ed25519) - possible?

I know the usual way of using getting shared secrets for encryption with ECC is DH, however, this only works with two keypairs of exactly the same kind, for example two curve25519- or two ...
1
vote
1answer
301 views

Use curve25519 for ElGamal crypto [duplicate]

DJB described curve25519 in his paper which can be found here (PDF). It seems that the main purpose was for Diffie-Hellman key exchange. I think this means that Discrete Log is supposed to be hard on ...
2
votes
1answer
453 views

inverse problem about scalar multiplication on elliptic curve

Let $E$ be an elliptic curve over a finite field $F_p$. Given $n$ be a positive integer and $Q$ be a point on $E$, assume that $Q=nP$, how can we find this $P$? We can assume that $n|p-1$. If $n$ is ...
2
votes
1answer
220 views

Is this ECC encryption key sharing method okay?

Is this encryption key sharing okay to use? Or is much better to use ECIES? $G$ = base point $a$ = Alice’s private key $b$ = Bob’s private key $A = aG$ = Alice’s public key $B = bG$ = Bob’s public ...
9
votes
3answers
1k 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 Castle, for example, ...
1
vote
3answers
396 views

Point decompression on an elliptic curve

I'm programming an elliptic curve cryptosystem and I'm having difficulty with decompressing points. The following information is from my project specification as to my understanding: Given a point ...
4
votes
1answer
725 views

ECC vs RSA: how to compare key sizes?

I know and I have understood the details of RSA, elliptic curve cryptography, (EC)DH and (EC)DSA. I keep reading everywhere that (if we don't consider non-deterministic computers) "ECC can achieve ...
0
votes
1answer
73 views

What is more efficient, pairing based cryptography or non pairing based cryptography? [closed]

As far as I know non pairing pairing based cryptography is less time consuming than pairing based because, pairing based uses complex operations. Are there any advantages of pairing based cryptography ...
3
votes
1answer
138 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?
3
votes
1answer
124 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 ...
4
votes
1answer
531 views

Blockwise Montgomery multiplication

I have to implement a 256*256 bit Montgomery multiplier for pairing computations. The straightforward approach is to use a bit-serial version, but I would like to utilize the built-in 64*64 bits ...
1
vote
1answer
195 views

Simplified Example of ECC to use in the classroom

I have come up with the following rudimentary example of how ECC relates to asymmetric keys. Is this a valid explanation of ECC and its relationship to asymmetry? To only be deciphered by the person ...
2
votes
2answers
361 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. $2G$,$3G$,.....,$(N-1)G$ ...
0
votes
1answer
117 views

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 ...
3
votes
1answer
165 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. ...
1
vote
1answer
335 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 ...
3
votes
2answers
189 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 ...
1
vote
1answer
145 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 ...
0
votes
1answer
140 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. ...
1
vote
2answers
154 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 ...
6
votes
1answer
224 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 ...
1
vote
1answer
540 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 ...
2
votes
1answer
70 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 ...
6
votes
1answer
93 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 ...
5
votes
3answers
3k 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 ...
1
vote
0answers
55 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 = ...
1
vote
2answers
373 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 ...
11
votes
1answer
249 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), ...
2
votes
2answers
92 views

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? ...
4
votes
1answer
220 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 ...
1
vote
2answers
131 views

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 ...
2
votes
2answers
364 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 ...
3
votes
1answer
96 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 ...
3
votes
1answer
92 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 ...