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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1answer
547 views

Is there a flaw in this ECC blind signature scheme?

Recently I've found the following work on the internet: An ECC-Based Blind Signature Scheme The paper claims to be an ECDSA blind signature however it seems that their scheme has a flaw in it. The ...
7
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1answer
94 views

Safe generation of $k$ points on a curve such that the mutual discrete logs are hard?

I have a multiplicative group $G$ of prime order $p$ implemented using a twisted Edwards curve (similar to Ed25519). I want to compute a set of $k$ distinct points $P_1,...,P_k$ that generate $G$, ...
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2answers
370 views

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

The Problem is as follows: ...
4
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2answers
214 views

In Elliptic Curve, what does the point at infinity look like?

We know that for each point $P$ in curve $E$ there exists a minimum scalar $k$ such that $k*P$ equals the point at infinity. And the book Cryptography Theory and Practice by Douglas R. Stinson only ...
2
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1answer
199 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 ...
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1answer
35 views

Is there a way to project one elliptic-curve element to a subgroup with certain size?

For discrete logarithm we can pick a random number $n$ and project it to a subgroup. E.g. given a prime $p$ with $p-1 = 2\cdot a \cdot b +1$ we can compute $n^{((p-1)/a)} \equiv n_a \mod p$ after this ...
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0answers
59 views

ECC with 512bit compatible curves

I understand that given solutions for solving a discrete logarithm problem are on the order of 𝑂(2𝑛/2), ergo, 256bit private keys based on 25519 or secp256k1 have an effective bit strength of ...
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37 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
85 views

Authentication protocol for communication with Arduino Uno

I am using an ECDH key exchange to establish a shared secret between an Arduino Uno and an Android device. For this purpose I am using this library and more specifically Curve25519. This is the ...
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1answer
115 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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0answers
29 views

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
110 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
93 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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1answer
463 views

Why does Smart's attack only work on anomalous curves?

Nigel Smart's attack solves the discrete logarithm problem in linear time. It requires the curve, however, to be anomalous, i.e. to have a trace of Frobenius equal to one or, equivalently, to be of ...
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2answers
387 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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1answer
57 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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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 ...
22
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1answer
7k views

Why do the elliptic curves recommended by NIST use 521 bits rather than 512?

Wikipedia says in reference to the elliptic curves officially recommended by NIST in FIPS 186-3: Five prime fields for certain primes p of sizes 192, 224, 256, 384, and 521 bits. For each of the ...
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1answer
46 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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0answers
32 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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133 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 ...
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 ...
4
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2answers
611 views

Using Lattice-based cryptography for TLS\SSL

Given the general benefits of Lattice-based cryptography, such as: Post quantum Security Security from worst case scenario Efficiency What could the outlook of shifting from RSA \ ECC-based ...
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1answer
44 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 ...
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1answer
86 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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2answers
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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
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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1answer
54 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 ...
0
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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 ...
3
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0answers
74 views

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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104 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
522 views

Which one is faster GCM AES-128 or Curve25519?

As I know the symmetric encryption algorithms are faster than asymmetric encryption algorithms. But when I test GCM AES-128 and Curve25519 encryption time, I find Curve25519 is faster than GCM AES-128....
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41 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 ...
2
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2answers
197 views

Can you compress an elliptic curve private key in half?

According to this, a $n$-bit key offers about $n/2$ bits of security. That got me wondering, can you compress the key in half? At first blush, no, because the key is essentially a random number, and ...
3
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1answer
117 views

How to find q in a corrupted ECDSA signauture

I'm doing exam papers for a crypto exam and I've been given a corrupted ECDSA signature. I'm asked to check if the signature is valid (which I know how to do), however, the q value is corrupted and so ...
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0answers
88 views

Caculated time One Point Multiplication with double and method

I am using double and add method for point multiplication in affine coordinates. How we compute 1PM in double and method? Di Wang said one point multiplication consists of repeated addition and ...
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1answer
40 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 ...
2
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0answers
37 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 ...
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1answer
93 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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1answer
102 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
52 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 ...
2
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1answer
43 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 ...
2
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1answer
145 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
1k views

Performance of ECDSA, ECKCDSA and ECGDSA

It is proven that ECDSA algorithms are faster in key and signature generation compared to RSA. In addition, the signatures are much shorter. However, I would like to know the performance difference ...
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1answer
134 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 ...
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1answer
121 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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1answer
49 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 ...
3
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1answer
161 views

Edwards / Montgomery ECC over binary extension fields

I recentely had a discussion about the redesign of our ECC code for the library I'm collaborating on and the person I was discussing with came up with Edwards and Montgomery curves over binary ...
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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. ...
14
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2answers
2k views

Schnorr signatures: multisignature support

Schnorr signature is mentioned as a promising upgrade to bitcoin to improve scalability. It support multisignature, several signatures can be aggregated into a single, new signature. But I fail to ...