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.
1,150 questions
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Elipstic curves on finite fields
I've been reading: https://github.com/bellaj/Blockchain/blob/6bffb47afae6a2a70903a26d215484cf8ff03859/ecdsa_bitcoin.pdf
On page 22 it shows an eliptic curve over F17.
I have added the orange lines ...
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43 views
Sharing secret content with multiple recipients
I have a sender and N recipients, and am thinking of using the following scheme to send secret content to those recipients. This is similar context to a group chat or email.
I am no expert in crypto ...
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1answer
75 views
What is the difference between subtracting the modulus from a scalar field element and reducing it?
When implementing a Field element, we define the necessary operations on the data structure.
One function that I see is a "scalar reduce" function, which effectively reduces a random scalar so that ...
2
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1answer
60 views
Prove I know a value $v$ in a Pedersen Commitment without revealing it
Given a Pedersen Commitment:
$P = aG + vH$
Where $G$ and $H$ are points in some group. $a$ is a blinding value/mask and $v$ is the value I wish to commit to.
Is there a way to prove I know $v$ ...
2
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2answers
96 views
How to model a miniature elliptic curve?
For educational purposes I would need to work on an elliptic curve that has a small field, but holds the safety futures of a real curve. Is that possible to have such a curve!?
For example for the ...
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1answer
56 views
Why only non-prime order fields have small subgroup attacks?
Why don't prime-order curves have small subgroup attacks?
It seems that I can choose a Generator such that it has a small order, maybe 2 points, and so an attack could generate all of the points in ...
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1answer
38 views
What type of curves have co-factors?
I read that non-primes only have co-factors, but Edwards have a co-factor and it is defined over Fp s.t. P = 2^255 -19 which is prime right?
How is the co-factor created, some have co-factor 4 and ...
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1answer
40 views
For an elliptic curve, what is the difference between the base field modulus $Q$ and subgroup $r$
What is the difference between the basefield modulus $Q$ and a subgroup of prime order $r$?
They are all fields, but what is their relevance to the curve they are defined upon?
How does this relate ...
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1answer
54 views
How does Diffie Hellman protocol work in Bitcoin Blockchain Transactions?
Greetings to all! Please explain how the Diffie-Hellman protocol works in Bitcoin?
That is, in Blockchain Transactions, there is also a total number of "K" recipient and sender?
"K" the recipient and ...
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9 views
Are compressed pairings used for Barreto-Naehrig curves in practice?
In 2009 Galbraith and Lin wrote the article "Computing Pairings Using x-Coordinates Only" https://link.springer.com/article/10.1007/s10623-008-9233-3, where they proposed to compute pairings on ...
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22 views
What books should I read as an introduction to elliptic curves? [closed]
Current Knowledge:
Studied Set theory in computer science
Basic undergraduate mathematics
Basic understanding of Fields and Number theory.
Goal:
To be able to code Elliptic curves from scratch, ...
2
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1answer
45 views
In ECC scalar multiplication, how do `add(Q, Q)` exceptions occur?
Consider some scalar multiplication algorithm for a prime order (Weierstrass) curve $E$ with order $\ell$. Bernstein and Lange's SafeCurves: Completeness page mentions:
The problem is that the ...
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1answer
136 views
+100
Proving that the least significant bit of an elliptic curve discrete logarithm is $0$
Suppose I have a secret value $a$ which maps to a public point on an elliptic curve $A = a \cdot G$, where $G$ is a generator of the elliptic curve of primer order $q$.
Can I prove to someone that ...
2
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1answer
42 views
Embedding POW in an EC public key
I have an application where it would be advantageous for an attacker to have to spend a long time generating public keys.
To do this, I require that the hash of the public key be less than a certain ...
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2answers
512 views
Geometric interpretation of an Edwards curve
Addition on an elliptic curve in Weierstrass form (over the rationals) is typically depicted with the following figure:
(Image CC SA 3.0 https://en.wikipedia.org/wiki/File:ECClines.svg)
To add two ...
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1answer
57 views
How to compare the order of elements in cyclic groups?
In a cyclic group with randomly looking behavior like the one used in secp256k1, is there any known efficient algorithm to compare the order of two randomly given elements $P_1$ and $P_2$ and find out ...
2
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1answer
97 views
What is the point at infinity on secp256k1 and how to calculate it?
I hear that there should be a point at infinity on secp256k1. I wounder how to calculate it and what does it even mean. I tried to calculate it as $P_{inf}=P+(-P)$ but this gives different results for ...
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1answer
60 views
Does Elliptic Curve Integrated Encryption Scheme (ECIES) provide IND-CCA2 security?
I am looking for a faster alternative to RSA with OAEP as a IND-CCA2 public key scheme. Elliptic Curve Integrated Encryption Scheme might be a candidate, but I am not sure if it provides IND-CCA2 ...
7
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2answers
396 views
Should we use IANA groups 14 (MODP), 25, and 26 (ECP)?
By looking at SonicWall Knowledge Base article Key exchange (DH) Groups Supported - Site to Site VPN:
It appears that our firewall supports DH group 25, and 26. Almost everywhere I've seen, they've ...
5
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1answer
105 views
Are EC public key values evenly distributed?
At my Day Job(tm) we've encountered a bug wherein if the leading digit of the X or Y value of a public key are zero, "shit happens" (this bug is in our code - I'm not suggesting there's some problem ...
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1answer
73 views
Could the reversibility of adding function be considered a weakness to secp256k1?
I have just started studying cryptography and secp256k1. I just wonder that adding two points can be easily reversible when the generator point is publicly known.
I mean that if $Q=\operatorname{...
2
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1answer
57 views
Why SM2 ECC parameters does not specify cofactor h?
Recently I've been studying the ECC with the Chinese SM2 standard. One question is on standard part 5, parameters definition, it only defines $p, a, b, n, XG,$ and $YG$, but not cofactor $h$.
I found ...
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28 views
What is connection between curve used for ec_keypair generation and ClientHello extension?
Im currently experimenting setting up my first SSL server and don't understand the following scenario:
My server has a generated ECC keypair with group identifier $A$ (e.g Brainpool256r1).
During TLS ...
6
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1answer
187 views
Point halving on elliptic curves of even order
I am trying to understand how point halving on elliptic curves of even order works. Specifically: suppose $g$ is an elliptic curve, and $G$ is a generator point on this curve. The order of group ...
2
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61 views
Proving that a point on elliptic curve is smaller than half of group's order
Let's say I have an elliptic curve where generator $G_1$ has prime order $q$. Let's also say I have committed to a point $A_1 = a \cdot G_1$. Could I use the scheme below to prove that $a < \frac{q}...
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44 views
Is it necessary to transmit two or three points of an elliptic curve?
Are there cryptographic protocols, where a party should transmit by communication channel simultaneously two or three $\mathbb{F}_q$-points of an elliptic curve over a finite field $\mathbb{F}_q$? ...
2
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1answer
87 views
Proving that two points on elliptic curve are within range
Is it possible to prove that a point on an elliptic curve falls within a given range of another point, without revealing the distance between them. For example:
Let's say $X$ and $Y$ are two points ...
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2answers
68 views
Does this pairing-based signature scheme work?
Suppose $g$ is a pairing-friendly elliptic curve with subgroup generators $G_1$ and $G_2$. Suppose also that $M$ is the message I want to sign.
Setup
Compute $A = a \cdot G_1$ and $P = p \cdot G_2$, ...
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0answers
49 views
Password recovery on Block-chain ( Hyperledger sawtooth in this case)
I am using bip39 specifications for 24-word mnemonic, Whenever user registers with his/her password, A Scrypt key is generated from this password and used to encrypt this mnemonic, The admin of the ...
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44 views
Proving key equivalence across different elliptic curves
We can use the technique described in this answer to prove key equivalence across two elliptic curves of different order. I'm wondering if modifying the technique as described below would compromise ...
2
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1answer
61 views
Does this simple signature scheme work?
Let's say my public key is defined as $P = p \cdot G$, where $p$ is my private key and $G$ is a generator point of an elliptic curve. If I wanted to sign a message $m$, could I do the following?
Hash ...
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1answer
44 views
Can a TLS certificate using ECC secp384r1 as PK algorithm uses RSA for signature
If a TLS certificate public-key algorithm is ECC secp384r1 or ECC prime256v1, is it possible to have RSA as a signature ...
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1answer
82 views
What is the recommended minimum key length for ECDSA signature
I want to identify the proportion of certificates that use unrecommend ECDSA key length for TLS certificates based on some data I collected.
By looking at a standard like NIST for example, I find ...
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1answer
155 views
How to send a secure public message using elliptic curve?
With ECDSA signature anyone can be sure if I own the private key or not without revealing it...
But if I have a website, and the users want to send me a secret message with this website using the ...
2
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2answers
58 views
Rational exponents on group generators
In elementary concepts, mostly scalar exponents shows up in group operations:
$g^x$
As one may encounter in more advanced papers, there are rational exponents over generators. Simply seems like:
$g^...
2
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2answers
83 views
Is it secure to encrypt a 1 GB data file with AES-CTR by ECIES?
I have a file system with share resources over the network (like a shared FTP)
Is it possible to use the client's ECC public key to encrypt the AES key, and then use the AES in CRT mode for the file ...
2
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1answer
64 views
Can we use PHE or SWHE instead of bilinear pairings in ZK-SNARKS?
In ZK Snarks bilinear pairings are used to do "encrypted computation". I was wondering if we can use Partial Homomorphic Encryption or Somewhat Homomorphic Encryption instead of bilinear pairings. Can ...
3
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1answer
114 views
Do I need to provide entropy to secp256k1_ecdsa_sign() ?
using secp256k1_ecdsa_sign() I noticed the same data signed multiple times, coming back with the same signature. I always thought that signatures are different because random data is somehow involved....
3
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2answers
81 views
Why ECDSA has its form?
According to Wikipedia, if Alice wants to sign some message, she computes $s = k^{-1} (z + r d_A)$ then sends $(r, s)$ to Bob.
I don't understand why they use this particular formula $s = k^{-1} (z + ...
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1answer
42 views
How to name a signature algorithm that has no hashed component
I have an API-based service that digitally signs messages with RSA or ECDSA (system decides during runtime). The input is base64 encoded SHA-256 hash and the output is:
signature of this hash
...
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2answers
100 views
What risks are involved when choosing a curve for an implementation of ECC?
What risks are involved when specific curves are chosen for an ECC implementation?
How should I audit a system that uses ECC with regards to a specific curve?
Bonus: How can I (or a cryptographer) ...
6
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1answer
102 views
Why would the use of Curve25519 in Dragonfly leak information?
An answer explaining Dragonfly, a form of key exchange used in WPA3, has an interesting footnote:
One final note: reviewing the Firefly RFC, I see that it would (as written) leak some information ...
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1answer
37 views
ECDSA Signing and verifying signatures between Python and JS [closed]
I create a signature on js, here jsrsasign
Signatures are obtained in the format:
...
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1answer
54 views
Equivalent RSA modulus for NIST P-192 and P-521 elliptic curves [duplicate]
At www.keylength.com, I found the following table of ECC field size and the corresponding RSA modulus recommended by NIST.
ECC Modulus RSA Prime Size
160 1024
224 2048
...
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1answer
162 views
How does the process of creating a new secure Elliptic Curve look like?
I'm especially curious about the technique djb would have used to come up with his Curve 25519. Say I have already written down my goals, such as - Twist Secure, Speed, Side Channel resistance, etc. ...
2
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3answers
60 views
For elliptic-curve cryptography, does a 256-bit key imply that $x$ and $y$ are each 256-bits or 128-bits?
In the wikipedia article, the claim is made that "256-bit elliptic curve public key should provide comparable security to a 3072-bit RSA public key". Since, in ECC, the public-key consists of a point $...
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35 views
Algorithm to compute DLOG for elliptic curve $E(F_p)$ with order p
I was reading about elliptic curves in this pdf. Page 55 of the pdf states that if number of points on elliptic curve #$E(F_p) = p$, then there exists a p-adic logarithmic map that homomorphically ...
3
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0answers
54 views
What is frobenius map of an Elliptic Curve?
I was reading about elliptic curves from this PDF. Page 44 defines Frobenius map. It defines the frobenius map as $f(x,y) = (x^p, y^p) \bmod p$. Isn't it just an identity map? What's the use of this ...
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44 views
NIST ECC Curves without pairings
NIST FIPS.186-4 has standardized 5 ECC curves on field $\mathbb{F}_p$ (P-192, P-224, P-256, P-384, P-521) and 10 elliptic curves on binary fields. I need to use ECC curves without pairings for my ...
