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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6
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1answer
174 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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Elliptic curve and embedding degree

I am new to ECC. I am confused about what the embedding degree in an elliptic curve group represents and what is the impact of its values on the curve and security (small values or large values?) ...
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ECDSA, EdDSA and ed25519 relationship / compatibility

I'm trying to understand the relationship between those three signature schemes (ECDSA, EdDSA and ed25519) and mainly, to what degree are they mutually compatible in the sense of key pair derivation, ...
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Why Elliptic Curves?

What is the benefit of using elliptic curves over the standard finite field, when the cyclic subgroup we consider of the EC's solution group is just isomorphic to some integer residue class of prime ...
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1answer
1k views

Families of public/private keys in elliptic curve cryptography

I'm looking for a related key scheme for elliptic curve cryptography. The basic idea would be that there would be a master public key and a master private key. From the master public key, you could ...
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2answers
3k views

Can curve25519 keys be used with ed25519 keys?

Can curve25519 keys be used with ed25519? I'd prefer to use ed25519, but there isn't a fast java version. For my application, I'd like to use curve25519 until I can get a faster ed25519 for java. ...
7
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1answer
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Elliptic curve cryptography related key attacks [closed]

This question is an extension of Families of public/private keys in elliptic curve cryptography As described above, bitcoin "type 2" deterministic wallets use a root private/public key pair, where ...
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2answers
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Why is Diffie Hellman used alongside public keys?

I just read this post here: Why do we need asymmetric algorithms for key exchange? that asserts that in public key cryptography when asymmetric keys are used to secure communications, for parties to ...
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1answer
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Does Curve25519 only provide 112 bit security?

In a recent mail on the IETF CFRG mailing list it was claimed that: The (currently missing) security considerations (or somewhere) should describe why Curve25519 is ok when used in contexts where ...
9
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1answer
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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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Subtracting a point in elliptic curve cryptography?

I've had lots of practice adding points for my crypto class. However I've run into a situation where I need to subtract two points for decryption: ...
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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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Base point in Ed25519?

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 $(x, 4/5)\in E$ for ...
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2answers
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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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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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How to derive a symmetric key from ECDH shared secret?

I am trying to implement the internal primitives of ECDH. Currently I'm able to multiply the receiver's public EC point with the sender's private key to arrive at the shared EC point. Next step is to ...
4
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1answer
760 views

ECC - ElGamal with Montgomery or Edwards type curves (curve25519, ed25519) - possible?

I know the usual way of using getting shared secrets for encryption with ECC is DH, however, this only works with two keypairs of exactly the same kind, for example two curve25519- or two p256-...
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1answer
425 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 ...
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1answer
271 views

ECIES for not shared encryption

I am using ECIES (Cryptobox as poncho pointed out) for public key encryption with ECC between Alice and Bob with secret/public keys $(a,A)$ and $(b,B)$ respectively. So Alice encrypts to Bob with $$ ...
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2answers
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Different ways/algorithms for implementing AES

I have seen a couple software implementations of the Advanced Encryption Standard. They are pretty much straight forward, i.e. they are implemented exactly the same way as the AES is described. This ...
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1answer
534 views

ElGamal with elliptic curves II

There is an encryption scheme using elliptic curves given by @tylo explained here: @tylo's answer on ElGamal with elliptic curves and here: ElGamal with elliptic curves I. The encryption idea is to ...
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3answers
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Why is it impossible to find the private key from the public key?

From what I understand, based on the article A Beginner’s Guide: Private and Public Key Cryptography Deciphered, the public key is generated by performing $n$ tangent plus mirroring operations from ...
3
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2answers
195 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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1answer
752 views

ECC partially blind signature scheme verification

Continued from Is there a flaw in this ECC blind signature scheme? The problem I needed a partially blind signature scheme for one of my projects, but couldn't find one on the internet, so I've made ...
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2answers
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Why Smart's attack doesn't work on this ECDLP?

The Problem is as follows: ...
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3answers
298 views

Which area of Maths should I pursue?

I would like to know which area of Mathematics would be most beneficial to cryptography. Surely Algebraic Number Theory and maybe to a lesser extend, Elliptic Curves, are closely linked to ...
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1answer
488 views

Choice of ECC curve on USB token

I would like to get a USB token to secure my keys. My use case is protection of 3 GnuPG keys that I will be using 10 times per day at least. I plan to create a new key ring from scratch. Because ECC ...
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1answer
300 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 ...
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2answers
952 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 ...
6
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1answer
240 views

Are there groups where the computational Diffie Hellman problem is easy but the discrete log problem is hard?

I know that there are elliptic curve groups, used in pairing-based cryptography, where the decisional Diffie Hellman problem (ie. given $g$, $g^a$, $g^b$ and $c$, determine if $c = g^{ab}$ is easy but ...
6
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1answer
2k views

How to represent point-at-infinity in affine coordinate

In projective coordinates point-at-infinity can be identified with z=0. How to identify the point-at-infinity in affine coordinate. Whether x=0 and y=0 can be considered as point-at-infinity in ...
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3answers
327 views

Zero-knowledge transfer of value protocol inspired by EC El Gamal

This is a follow up on the question I asked here. I designed a scheme that allows the following: Alice has a value $a$ which she wants to keep secret Bob has a value $b$ which he wants to keep secret ...
3
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1answer
161 views

Attacks on schemes based on elliptic curves when the transmitted points are not on the curve

Some elliptic curve schemes require to send a curve point during the normal execution of the protocol. For example, ElGamal encryption and ElGamal signature require this. On the other hand, ECDSA does ...
3
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1answer
327 views

Is a hash function used to expand the key after ECC shared secret is complete?

I have designed an ECC engine in silicon that handles any curve in the form of $$y^2 = [ax^3 + bx^2 +cx + d] \mod(P)$$ The shared secret is then passed to symmetric encryption engine, which happens ...
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2answers
377 views

Roots in modulo field

I have a point $(X,Y)$ on an elliptical curve $E(a,b)$ where $a=-3$ and $B$ is a large number that is in hexadecimal from -51BD. To compress this point oficially in a program, we know that every $X$ ...
2
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1answer
798 views

Point at infinity for Jacobian coordinates

Why is the point at infinity ∞ in affine space corresponding to (1 : 1 : 0) for Jacobian coordinates? How can this be shown? Is ist just to add the negative and see that the outcome is 1:1:0 or is ...
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0answers
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Is my way safe to remove SSL CA Cert by DHT and PoW NodeID for a decentralized system?

To implement a decentralized system, I wrote a TLS like P2P net stack. The main idea is removing CA Cert from the whole system by using a DHT for Naming and Key Exchange. I am not a crypto expert, so ...
2
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2answers
636 views

64 bit Elliptic Curve key?

For a simple proof of concept project i'm (attempting!) to do, i've started looking into openSSL elliptic curve cryptography. However instead of the standard key lengths, 160-512. I'm interested in ...
2
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1answer
750 views

ElGamal with elliptic curves I

It is very interesting to see @tylo's answer on ElGamal with elliptic curves. Instead of mapping the message to the elliptic curve point it just reduces an elliptic curve point to its $x$ coodrinate. ...
2
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0answers
170 views

Finding the largest gap between the x coordinates of all points on an elliptic curve

Till now all we know is Hasse's theorem, which states that $|\#E(p)-(p+1)| \leq 2\sqrt{p}$, where $\#E(p)$ is the total number of points in $E_p(a,b)$. Is there any other theorem which defines ...
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1answer
395 views

hashing points of elliptic curves

Can I create a secure hash function $h: E(\mathbb{F}_p) \rightarrow \mathbb{Z}_q$ (for some $q$) where $E(\mathbb{F}_p)$ is an elliptic curve on the finite field of $p$ elements? By secure hash ...
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0answers
123 views

Implementing key mapping across different elliptic curves

From this answer I understood how to prove key equivalence across two elliptic curves. Now, I'm trying to figure out some more practical aspects of implementing this. Before jumping into questions, ...
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1answer
180 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 ...
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0answers
104 views

Using division polynomials to prove that EC discrete log is even

This question is related to the other question I recently asked. I'm trying to figure out if it is possible to use division polynomials to prove that knowing $A = a \cdot G$ we can prove that $a$ is ...
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1answer
167 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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2answers
92 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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3answers
12k views

Why Curve25519 for encryption but Ed25519 for signatures?

NaCl and libsodium libraries use Curve25519 for authenticated encryption (actually for sharing a key which is used for encryption) and Ed25519 for signatures. What is the purpose of using different ...
38
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2answers
22k views

ECDSA vs ECIES vs ECDH

Recently I started studying Elliptic Curve Cryptography and I just loved it. I want to transfer some big data (like 3KB), What is the best method, ECDSA, ECIES, or ECDH (and why)? I am confused, how ...
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1answer
8k views

How does ECDH arrive on a shared secret?

I read a brilliant, three part article on Elliptic Curve cryptography (one, two, three). It was able to explain Elliptic Curves to me in a way that didn't require a math degree to understand. The ...
24
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1answer
17k views

How strong is the ECDSA algorithm?

Some cryptographic algorithms are as strong as the size of their key is, while other have some weaknesses that limit their strength (such as SHA-1). How strong is the ECDSA algorithm, and does that ...