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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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1answer
66 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
62 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
31 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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0answers
76 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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1answer
543 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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46 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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2answers
200 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 ...
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1answer
127 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
91 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 ...
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45 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
138 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
118 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
73 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
45 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
177 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
214 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
158 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
50 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
181 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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39 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. ...
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2answers
938 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
123 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
97 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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3answers
650 views

Pedersen commitment in elliptic curves

I try to understand Pedersen commitment in elliptic curves over finite fields. I could use some clarification. Let's say we have two generators $G$ and $H$. Is that required that $G$ and $H$ are ...
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1answer
202 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 ...
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1answer
82 views

Can I use NaCl's scalar multiplication functions for Diffie Hellman Key Agreement?

I want to create some software that performs diffie hellman group key agreement but I don't want to reinvent the wheel even if I know how it's done. So I came accross the NaCl library, especially the <...
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1answer
541 views

ECDH security when no KDF is used

Let's suppose our device performs ECDH with a fixed, unknown, private key $\text{prv}$. It accepts as input any point $Q$ lying in the proper subgroup of the proper elliptic curve, then computes: $P =...
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1answer
99 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
849 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
1k 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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4answers
3k views

EC Schnorr signature: multiple standard?

I'm working on some EC-Schnorr signature code. Reading various papers on that, it seems EC-Schnorr is not standardized as well as ECDSA. For example, I found two main differences in two main actors ...
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1answer
524 views

Can Montgomery ladder multiplication be used with secp256k1?

While reading about Elliptical Curves and ECDSA, I found a paper ECDSA Security in Bitcoin and Ethereum: a Research Survey by Hartwig Mayer. On page 6, the authors say: The curve secp256k1 does not ...
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1answer
165 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 - ...
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1answer
1k views

Fast modular reduction

I am looking at ways to speed up modular reduction for the polynomial $$2^{256}-2^{32}-2^9-2^8-2^7-2^6-2^4-1$$ I have read the paper "Generalized Mersenne numbers" by J.A. Solinas, but it does not ...
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1answer
131 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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1answer
99 views

Generating a small EDDSA curve

I have an application that would benefit from very small (e.g. 16-20 byte) EDDSA keys and small signatures. It's an application where the goal is more to deter DOS attacks than "hard" security, so ...
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2answers
365 views

How to securely map messages to points on an elliptic curve

I'm implementing a demonstration hybrid cryptosystem in Python (FinCrypt, I know the name is bad) and I'm migrating over from my Weierstrass curve implementation, which was based off of this, to one ...
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0answers
55 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
34 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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2answers
298 views

Safe and computationally efficient way to verify a curve25519 identity?

A client identifies itself as a curve25519 public key. The server wants to verify the client owns the associated private key. Is there a safe and computationally efficient way of doing so? Which ...
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0answers
69 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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1answer
345 views

Complex Numbers on Elliptic Curves & Usage in Tate Pairing

I'm working with understanding the internals of the Tate Pairing. I was going through an example of the curve $E: y^2 = x^3 + 3x$ over $\mathbb{F_{11}}$. The author is showing the computation of $e(P,...
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2answers
347 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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1answer
145 views

A method for creating distributed public key for ECDSA, what are the risk factors?

There is quite a bit of literature on distributed ECC signing without a trusted dealer. Published works are mostly overly complicated, so I am proposing this simple technique which I am sure if it is ...
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1answer
778 views

What are the fastest attacks on ECDLP?

Consider the ECDSA protocol, which is applied in different environments e.g. the Bitcoin system (for user addresses, and transaction signing). What are the greatest threats in terms of algorithms ...
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1answer
335 views

Why is there no 'ECDSA' version of 'DHE-RSA-CHACHA20-POLY1305'?

So I was just checking my TLS cipherlist and noticed that there was a 'DSS' / DSA / ECDSA version of every ...
2
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1answer
142 views

URN for ECDSA signature algorithms (including hash algorithm)

The URN form ECDSA signature algorithm is urn:nist-gov:ecdsa. But I am not able to find a named URN for algorithm SHA1withECDSA or SHA256withECDSA. Up to now I've ...
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1answer
9k views

How does encryption work in elliptic curve cryptography?

So I think I understand a good amount of the theory behind elliptic curve cryptography, however I am slightly unclear on how exactly a message in encrypted and then how is it decrypted. So my ...