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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2answers
327 views

Why Smart's attack doesn't work on this ECDLP?

The Problem is as follows: ...
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0answers
36 views

Why do they use elliptic curve instead of circle or other simpler curves? [duplicate]

I am curious why people use elliptic curve in cryptography. I know the main requirement is DLP, but elliptic curve is not the only curve with such property. Some of curves seem to be even simpler. As ...
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1answer
96 views

Why are singular “elliptic” curves bad for crypto?

Consider the algebraic curve given by a short Weierstraß equation $y^2=x^3+ax+b$. If $4a^3+27b^2=0$, then there are repeated roots of the right-hand side $x^3+ax+b$. How are these repeated roots bad ...
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If curve bn256/bls12 support the isomorphism from $G_2$ to $G_1$?

Is bn256 or bls12 a type-2 pairing-friendly curve? As Dan Boneh said here While in many pairing instantiations this ψ exists naturally, in some instantiations it does not. However I can not find ...
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1answer
94 views

Are there any NIST curves with pairings?

NIST FIPS.186-4 has standardized 5 ECC curves on field 𝔽𝑝 (P-192, P-224, P-256, P-384, P-521) and 10 elliptic curves on binary fields. None of them seem to have pairings. Are there any standard ...
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1answer
85 views

Why is the strength of an Elliptic Curve Cryptography (ECC) half the size of the prime field size?

I've looked around and couldn't find a direct answer. As a general rule, I've read from various sources (here here, and here) that the strength of an elliptical curve key is half of the size of the ...
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3answers
169 views

Is the curve25519 algorithm a special(implementation) one of ECDH?

It's the first time for me to learn about Key Exchange Protocol. And I thought that in both ECDH and DH there is a necessary step to share some public infomation(the common parameters) to each sides ...
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0answers
31 views

short signature for EC

i'm building a low-power wireless sensor network in which each slave node has a public/private ECC key pair -- generated by the node itself during manufacturing.... the slave node is also provisioned ...
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1answer
45 views

How to calculate the order of the subgroup?

Given a curve with points over GF(p), a subgroup of prime order q and a co-factor h. How do I calculate the size of q which is ...
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1answer
53 views

Security strength EC signature for variable message size

I am implementing a system using some sort of 32bytes OTC (One-Time Code) and signing it with ECDSA to get verification of a public key owner. The key I'm using is ...
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1answer
27 views

Cost model for different curve models

Is there a cost model for each curve model and their conversions? For example: Take the curve models: Projective, Completed, Extended, Affine. Is there a table which shows how many multiplications, ...
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0answers
121 views

How does post quantum key exchange in OpenSSH 8 work?

OpenSSH 8 supports a post quantum KEX, namely sntrup4591761x25519-sha512@tinyssh.org It says in its description that it is basically NTRU + ECC X25519. However, I have tried but cannot understand how ...
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31 views

How to exchange a common key to a group of person using Elliptic-curve Diffie–Hellman (ECDH)? [duplicate]

In ECDH, when two persons want to share private keys, they first select a point $G$ on the elliptic curve and after that, each of them picks a random integer $a$ and $b$, respectively, and multiply ...
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2answers
80 views

How to find at least one private key from a large list of compressed public keys secp256k1

Not long ago I saw a discussion on the Bitcoin Talk forum: https://bitcointalk.org/index.php?topic=5060735.msg50736695#msg50736695 Please give advice and working methods? Is it possible to find at ...
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1answer
49 views

Check if two ed25519 points are equivalent

How to check if two points are equivalent given their projective coordinates (XYTZ)? For example if I do unproject() of a point to pass from XYTZ to XY coordinates and then I come back to XYTZ ...
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1answer
25 views

montgomery reduction multiplicative identity

How do you figure out the multiplicative and additive identity with respects to R? I pick some R such that gcd(R, N) = 1 where N is the size of the group. Given some field element x in the group, I ...
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1answer
55 views

Computing a sextic twist

Let $(x,y) \in E'_{\mathbb{F}_{p^2}}$ be a point of the sextic twist. I am currently trying to compute: $\psi : (x, y) \leftarrow (\mu^2x,\mu^3y)$ with $\mu \in \mathbb{F}_{p^{12}}$ the root of $(Y^...
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0answers
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Doubt in computing $g^\frac{1}{\delta+x}$ where $x \in \mathbb{Z}$

I was going through Zero Knowledge Set Membership and came across the following: Given $x \in \mathbb{Z}$ and $g$ is the generator of a multiplicative group $\mathbb{G}$ how do we compute $g^\frac{1}{...
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0answers
37 views

Regarding the isogeny path problem

Given two elliptic curves, it is hard to calculate an isogeny of large degree between them. Does this only apply to supersingular isogenies or to ordinary ones as well? Additionally, is the mapping ...
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1answer
39 views

ECDSA over $\mathbb{F}_{p^n}$ for $n>1$. How to calculate $r$ and $s$

I'm having some trouble understanding how to calculate $r$ and $s$ as specified in the wikipedia page for ECDSA (https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm) We can see ...
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1answer
78 views

Simple math ECDSA example

I'm trying to setup an ECDSA math example using just integer math and multiply (no EC). The purpose is just to help people understand why this works with out the added complication of understanding ...
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0answers
34 views

Security strength of JPBC Type A curve compared to SecP curve

I recently encountered some problems when learning about the JPBC library. Does the curve generated by (J)PBC using the method typeAcurvegenerator(160,512) and the ...
2
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1answer
62 views

Share secret non-interactively in a verifiable way

In an application, I need to share a secret (random number) with a Group of known receivers over a public channel (a Blockchain) but each receiver needs to be able to check that the others received ...
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0answers
102 views

How do pairings behave on G2/twist points off the prime order subgroup?

$\newcommand{\F}{\mathbb{F}}$ Consider the ate pairing defined on a curve $G_1 = E(\F_q)$ and $G_2 = E'(\F_{q^r})$ where $E'$ is a twist of $E$ with the twisting isomorphism defined over $\F_{q^r}$. ...
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1answer
98 views

Converting EdDSA Keys to EC-KCDSA keys

I'm trying to create BIP32 like key derivation for EKCDSA by riding over a BIP32-EdDSA derivation Can anyone tell me if there is a glaring problem with my conversion technique? ...
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1answer
47 views

Question on using endomorphism on secp256k1 and negative results

I have read section 3.5 (algorithm 3.7) in "Guide to Elliptic Curve Cryptography", and have been trying to implement endomorphism on secpt256k1 to speed up calculating $kP$ by changing it into 2 point ...
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1answer
42 views

Lenstra's ECM Algorithm - field requirement

In Lenstra's ECM algorithm, $\#E(\mathbb{F}_{p})$ is required to have small prime factors. Why is this so? I understand that the p-1 method is efficient for factoring N with small factors. The ECM ...
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1answer
109 views

Generating a random point on an elliptic curve over a finite field

I have coded an implementation of elliptic curves in order to apply some of the ECC algorithms. However, in most of them, Alice needs to choose a point P on a given curve. What is the general ...
2
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1answer
187 views

Can you tell me why doing scalar multiplication of a point on a Elliptic curve over a finite field gets to a point at infinity?

I am reading Programming Bitcoin. The author said: Another property of scalar multiplication is that at a certain multiple, we get to the point at infinity (remember, the point at infinity is the ...
2
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1answer
141 views

Why is encryption slower than decryption in elliptic curve cryptography (ECC)?

While performing encryption using public key and decryption using the private key, I am always finding that encryption takes more time than decryption in elliptic curve cryptography (ECC). It's the ...
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1answer
47 views

Book request about elliptic curves, RSA and DSA

I understand that this question can be hardly downvoted, but so be it if someone gives me really useful references :) I wanna learn difference (deeply) between RSA, DSA, and ECC, especially I am ...
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1answer
111 views

What are the inverse operations in elliptic curve cryptography?

Public-private key cryptography is based on inverse operations that use separate input. In elliptic curve cryptography, what are those inverse operations?
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0answers
37 views

Explanation of Gallant-Lambert-Vanstone method / Endomorphism speedups [duplicate]

Can someone explain how the Gallant-Lambert-Vanstone method works (or which literature explains it)? It is also unclear to me how the Frobenius endomorphism can be used in some cases for a speedup. ...
2
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1answer
93 views

Are there shortcuts for computing ECC Point multiplication?

I'm trying to learn about elliptic curve cryptography. Let's say you have point $P$ and 256 bit number $n$ and you want to compute $nP$. It sounds like computing additions one at a time is not ...
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1answer
51 views

How to multiply two Public Keys in Elliptic Curve in Go

I am working on a messaging client similar to Signal. I am stuck on implementing Tripartite Diffie-Hellman handshake in which three DH exchanges are combined to authenticate both parties and produce ...
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2answers
387 views

Does secp256k1 have any known weaknesses?

I am wondering whether there are any properties of the curve which would technically make it easier to attack than any other curves of 256 bits in size. I have heard that being a Koblitz curve, it ...
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1answer
43 views

Why public key has two parts in my secure messaging client similar to signal

I am working on a Golang code similar to Signal protocol. I need to modify it. I am confused on tripartite Diffie-Hellman handshake part of code, i.e. why public key has two separate parts as compared ...
3
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1answer
130 views

ECDSA signing process

I am trying to learn how ECDSA works. I do not have a background in maths, but have been following a guide which has built me up from finite fields, elliptic curves. I am unable to figure out how a ...
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1answer
76 views

Why is it better to add and double points on an elliptic curve using projective space?

I have been given a textbook which defines the addition of two points on an elliptic curve and the doubling of a point on an elliptic curve. This textbook explains elliptic curves in projective space ...
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1answer
78 views

Convert affine to projective coordinates and vice versa in ECC?

I am working on a small project. An elliptic curve equation with y^2=x^3-3x+27 mod 43, a point $Q=(1,38)$, using point doubling method https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication#...
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1answer
432 views

Curve25519 over Ed25519 for key exchange? Why?

I've been reading up on the Signal Protocol (in this PDF) and it seems to be using Curve25519 for ECDH and EdDSA (with Ed25519) for signatures. My question is why not use only Ed25519? This ...
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2answers
382 views

How is EC key encoded in PKCS#8?

I just started working with certificates and signatures. For an application I write I need a key pair for ECDSA signatures, using the elliptic curve secp384r1 (aka NIST P-384). I produced such a key ...
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1answer
106 views

How can I exploit the structure of the secp256k1 prime for fast arithmetic?

I'm implementing logic on an FPGA (programmable chip) that does the key verification part of ECDSA on the curve secpk256k1, in which all operations are mod p where $p = 2^{256} - 2^{32} - 2^9 - 2^8 - ...
4
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1answer
90 views

Curve25519's Y coordinate of Basepoint origin

The paper High-speed high-security signatures by Bernstein et al. introduces the Edwards curve Ed25519. Concerning the base point $B$, it says that $B$ is the unique point $(x,4/5)∈E$ for which $x$...
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0answers
51 views

Isogeny of elliptic curve

If we have two elliptic curves $E$ and $E'$ and the points of both elliptic curves are same. Then all the points of $E$ map to all the points of another elliptic curve $E'$. For example $E$ has ...
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0answers
26 views

How does the order of a group, it's torsion subgroup and the co-factor link?

Given an elliptic curve that defines some group of non-prime order, with co-factor h. Would it then have a h-torsion subgroup? What are the implications for ECC ...
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0answers
63 views

Deterministic generation of RSA keys for IPFS / OrbitDB [duplicate]

I am in the process of working on a decentralized application using IPFS and OrbitDB. IPFS uses 2048 bit RSA keys for the Node runtime peer-id and secp256k1 for read/write access in OrbitDB. For ...
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2answers
307 views

Difference on montgomery curve equation between EFD and RFC7748

There is a subtle difference between the 2 implementations for a Montgomery curve defined from the 2 following links https://hyperelliptic.org/EFD/g1p/auto-montgom-xz.html A = X2+Z2 AA = ...
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2answers
315 views

Can Shamir’s Trick crack the cryptographic strength of ECDSA?

Recently stumbled upon a discussion in the forum What is Shamir’s Trick used for? Are there any such examples?
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2answers
45 views

Elliptic curves - operations in larger groups - performance

According to my measurements and to this work, it seems that operations, for example scalar multiplication, are more expensive in larger groups. If I have, for example, an 80-bit elliptic curve and an ...