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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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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
138 views

Why do elliptic curves require fewer bits for the same security level?

I'm studying the basics of cryptography and I didn't understand why elliptic curves use fewer bits. For example, finite-field Diffie-Hellman needs at least 1024 bit and it's a DLP, but elliptic ...
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1answer
29 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
43 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
56 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
26 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 ...
4
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3answers
459 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 ...
2
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1answer
181 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
265 views

Can I use a random number generator with seed for key derivation?

I don't know a thing about cryptography so I'm judging by what I see in APIs docs. I want to use ECDSA or ECDH in my project. And what I know so far about key derivation in this systems is that they: ...
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1answer
57 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 <...
2
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1answer
371 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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0answers
34 views

Cryptography over non-elliptic curves [duplicate]

I'm just curious, what is the decisive factor to be restricted to elliptic curves? In theory, as long as we have a generator $g$ to have a suitable key pair $K, k: k\in F_q $ such that $K\equiv g^k \...
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0answers
41 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
34 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
155 views

Zero-knowledge transfer of value protocol II [closed]

This is an improvement of the protocol described here. The protocol does not require trusted setup and is very efficient (much more efficient than anything else I could find). The protocol allows the ...
3
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1answer
111 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
47 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 ...
5
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1answer
200 views

How can I generate a Koblitz curve?

Is there the way to generate new Koblitz curves, over $\mathbb F_{2^n}$ and $\mathbb F_p$? The Certicom SEC 2 standard says: The recommended parameters associated with a Koblitz curve were chosen ...
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4answers
2k 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 ...
2
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1answer
366 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 ...
3
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1answer
55 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 - ...
5
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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 ...
4
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1answer
54 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 $x$...
2
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1answer
75 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 ...
5
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2answers
178 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
44 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 ...
0
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0answers
22 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
247 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 ...
0
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0answers
47 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 ...
3
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1answer
297 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,...
5
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2answers
278 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
118 views

Is there a feasible way to generate an RSA key manually the same way as it is for an ECC one?

In elliptic curves, a private key is just a random number, and one relatively small compared to other crypto systems (256 bits for ECC vs 4096 bits for RSA for example). Suppose I don't trust ...
3
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1answer
113 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
471 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 ...
2
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1answer
303 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 ...
3
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1answer
116 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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3answers
4k views

How effective is quantum computing against elliptic curve cryptography?

I've been reading the Wikipedia page on Elliptic-Curve Cryptography and I came across the following. in August 2015, the NSA announced that it plans to replace Suite B with a new cipher suite due ...
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2answers
7k 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 ...
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0answers
53 views

Can we input $(0,0)$ in ECC addition?

If I have two points which are $(0,0)$ and $(6,9)$, then I apply addition $P + Q$ in ECC .Thus the answer give $(0,0)$ or $(6,9)$? Or is it not valid? Please explain, I'm confused.
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1answer
86 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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1answer
100 views

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

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

Why doesn't the formula work when checking two ECDSA signatures?

There are two generated ECDSA signatures X - Private key S = ((Z + (X * R)) / K) mod n S` = ((Z` + (X * R`)) / K`) mod n G - Base point, of order n; ...
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2answers
1k views

Can deterministic ECDSA be protected against fault attacks?

In a paper by Barenghi and Pelosi, it was described that fault attacks could be used to derive the secret key when using deterministic ECDSA as described in RFC6979 by @Thomas_Pornin Deterministic (...
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2answers
42 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 ...
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2answers
92 views

Which hash is used when providing signature algorithm ED25519 or ED448?

From TLS 1.3 there are two signature algorithms using edDSA: /* EdDSA algorithms */ ed25519(0x0807), ed448(0x0808), All the other signature-...
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0answers
58 views

ECDSA - why is the first part of the signature used in the second?

Using the terminology of https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm Why is the second part of the ECDSA signature defined as: $s = k^{-1}(z+rd_A)\text{ mod n}$ ...
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0answers
29 views

Why doesn't the JOSE suite/JWA include ECIES?

The JOSE suite specifics use of RSA-OAEP (for when one party has an RSA key) and ECDH (for when two parties have EC keys) in JWA. Why doesn't it include ECIES? It seems like a way to derive a key ...
2
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1answer
59 views

Secure multi-party computation for digital signature

Is there any practical algorithm that will allow to use public key cryptography (RSA or ECC) in the following way There are N parties. Up to M are malicious adversaries (were trusted, but got taken ...
2
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2answers
53 views

ECC How do the Curve, its number of bits of security and key size affect the maximum size of ciphertext?

First of all, excuse me if this question is too noobish. I'm trying to understand how these things are relate: Curve type, its 'number of bits of security', size of the key and the maximum ciphertext ...
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1answer
2k views

How to find the order of a generator on an elliptic curve?

I was looking out to find optimum generator for an elliptic curve $E$ over a prime field $\mathbb F_p$. I found the following algorithm: Choose random point $P$ on the curve. Find the order of a ...