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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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153 views

Elliptic 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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0answers
102 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
99 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 ...
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2answers
227 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 $v$ ...
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2answers
125 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
119 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 ...
3
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1answer
85 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
140 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
309 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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0answers
16 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 ...
2
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1answer
70 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 ...
7
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1answer
319 views

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 prime order $q$. Can I prove to someone that the ...
2
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1answer
59 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 ...
8
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2answers
628 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
90 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 ...
4
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1answer
616 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 ...
4
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1answer
163 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 ...
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2answers
1k 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 ...
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1answer
186 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
146 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
181 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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1answer
553 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 ...
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0answers
159 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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0answers
51 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
222 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
79 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
57 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
74 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
272 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
1k 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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2answers
354 views

How can I send a secure public message using ECC? [closed]

I want to know how to send a secure message protected with elliptic curve private and public keys.
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2answers
70 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
108 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
101 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
186 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....
6
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2answers
138 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
79 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 ...
3
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2answers
186 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
188 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
262 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: ...
0
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1answer
502 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
208 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
206 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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0answers
46 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 ...
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0answers
173 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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0answers
100 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 ...
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1answer
71 views

How do you perform the invert operation of Pm = kP on elliptic curve?

I'm stuck on this, may be due to the fact I'm missing something. If I'm right, in order to reduce a Message to a point on an elliptic curve the operation is: $\text{MsgPoint} = \text{msg}\cdot P$ ...
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2answers
254 views

Problem on Elliptic Curve Point Doubling

Given an elliptical curve e.g. from “Understanding Cryptography” by Parr & Pelzl §9.2 Example 9.5: $y^2 = x^3 + 2x + 2~~~~ mod~17$ And given a primitive $P = (5, 1)$, the book indicates: We ...
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0answers
73 views

Parity of the order of a element

Given an element $g$ in a cyclic group $G$ of known order $m$ its easy to test if $m$ has even or odd order. In other words $\textrm{ord}(g) \pmod 2$ can be computed easily. In some cases where the ...
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2answers
2k views

A private key with multiple public keys?

I'm trying to design a wallet, where any number of public keys can be handed out. Say Alice hands out the public keys to receive messages. She doesn't want others to be able to link all of the public ...