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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Why does Ed25519 scalar multiplication allow values larger than the subgroup order?

The GeScalarMultBase function is documented like so. From the way it is documented we see that it expects a little-endian value and has a precondition that constrains the range it accepts. ...
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Check validity of generated parameters for SIDH

In section 4.1 of the paper Towards Quantum-Resistant Cryptosystems From Supersingular Elliptic Curve Isogenies by Feo, Jao and Plût it is described how you generate valid parameters for the SIDH ...
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1answer
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The protocol of provisioning shared key to multiple devices?

I have a bunch of IoT devices which can do ECDH/ECDSA and AES. They all are connected to some server and know its public key. In turn, server knows each public key of the corresponding device. Devices ...
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196 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 ...
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1answer
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Get points of an Elliptic Curve defined over a Finite Field on Twisted Edwards Extended Coordinates

I'm working on a crypto library, and I need to perform some tests for the implementation of: Point Addition. Point Subtraction. Point Doubling. Scalar Mul Point. The operations are performed on ...
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Elliptic curves in Edwards form (or Edwards curve) and addition formulas

In recent studies on elliptic curve cryptography, Edwards curves are remarkable examples on this field. Studies show that this kind of elliptic curves provide faster computation compared to ...
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1answer
138 views

curve25519 by openSSL

How can i generate ec curve25519 keys using openSSL? When I run openssl ecparam -name curve25519 -genkey -noout -out private.ec.key I have this message ...
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1answer
38 views

Does any problem arise when the order of an elliptic curve is equal to its prime field modulus? [duplicate]

Regarding cryptographic schemes in elliptic curve cryptography, is there a problem with having the order of an elliptic curve being equal to its prime field modulus? That is, an elliptic curve where $...
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What is the cryptography involved in the initial setup of a cryptocurrency?

I keep hearing that when a cryptocurrency is created it goes through an initial setup phase wherein cryptographic parameters are created that are used by the cryptocurrency network throughout its ...
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Modifying Elliptic Curve Parameters

For context, I was watching this bit of the video: which goes over this source code. The piece is about elliptic curve cryptography and how it works. I want to use some of this knowledge to make my ...
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1answer
130 views

Testing PRNG quality from ECC public keys?

Having a large set of ECC public keys $P_i = n_iB$ on a fixed curve $E$ over a prime field, is there a way to determine if coefficients $n_i$ were generated using a bad PRNG? In other words, can a ...
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The ECC private key is generated with 0x00 at the beginning.(prefix)

I created a private key using the prime256v1 curve. My purpose is to get a 32 byte private key. However, the private key is preceded by 0x00, resulting in 33 bytes. Why is this happening? The only ...
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How does Diffie–Hellman differ from elliptic curve Diffie–Hellman?

I didn't understand how ECDH actually works. Disclaimer: I know very little about elliptic curves. Here is how DH works: Alice and Bob agree on a prime number $P$ and a generator $G$. (They use one ...
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How to get a random point of a specific EC group with cofactor Not-Equal 1?

We got a EC group generated with point G, and the cofactor of E(G) is with the similar size of the Order. Now we need a random point of E(G) and not revealing the "logarithm" of the random point, so ...
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1answer
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Are there any security risks in using Elliptic Curves defined over fields $\mathbf{F}_{p^n}$ where $n>1$

I've recently been studying elliptic curves, and I've found that most of the current implementations use fields $\mathbf{Z_p}$ or in some cases $\mathbf{F}_{2^n}$. All the reasons I've seen for not ...
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1answer
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Replacing elliptic curve diffie-hellman primitive with elliptic curve cofactor diffie-hellman for specifc curves?

From what I've read about elliptic curve Diffie-hellman with and without cofactor (I am pretty new to the whole thing so I am not able to understand everything) is that when the cofactor of the curve $...
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Half of any bitcoin (crypto) public key - (public key half) is possible

I found a topic on bitcointalk Public key x and y == Double(Half of the Public key x and y) half any public key is possible how that possible , in crypto there is subtraction and multiplication ...
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Why does Hasse's theorem sometimes seem to be invalid?

In order to generate secure elliptic curves, this answer recommends to Calculate the cardinal $|E(\mathbb{F}_p)|$ Check this cardinal is in the hasse interval (with $p$ prime) and to ...
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1answer
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Elliptic Curve Cryptography insecure when input does not lie on the curve?

I am new to Elliptic Curve Cryptography and I was reading up on it online when I came across this link. It stated the following. Unfortunately, there is a gap between ECDLP difficulty and ECC ...
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1answer
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Reasoning about WebCrypto ECDSA choices: P-256/384/521, SHA-1/256/384/512?

When implementing EC signing/verification in Javascript, the only options available via the WebCrypto API are: Curves: P-256, P-384, or P-521 Hashes: SHA-1, SHA-256, SHA-384, or SHA-512 If I was ...
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1answer
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Question regarding of the ECC test vector format

I am trying to find some ECC test vector for using. I just find some post (like this) and github resource (like this ) They are good reference to my C test code but I'd like to get some more advice... ...
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1answer
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How are Elliptic Curve private and public keys actually used to encrypt or sign data?

I've read article after article about curve parameters, generator points, the dot operation, and how you dot the generator point priv times to get the ...
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2answers
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Diffie-Hellman Primitives in SP800-56A

I wonder if someone can give an explain about the different between two Diffie-Hellman Primitives defined in SP800-56A, CH5.7.1 5.7.1.1 Finite Field Cryptography Diffie-Hellman (FFC DH) Primitive 5.7....
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1answer
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Valid private keys on curve25519

Given that valid private keys on curve25519 must be less than the order of the curve which is (as I understand) already smaller than 2^256, AND a valid key must be clamped to be divisible by 8 and ...
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Is Elliptic Curve Diffie-Hellman (ECDH) still secure if I use the public key more than one time?

Elliptic Curve Diffie-Hellman (ECDH) with Public parameters: Ep (a,b) and G = (x, y) Private Keys: ...
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1answer
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Is inversion always cheap with Twisted Edwards curves?

I'm reading on Jubjub, which is planned for the next upgrade of Zcash. It is based on a Twisted Edwards curve with parameters $a = -1$ and $d = −(10240/10241)$. The reading says Jubjub does not need ...
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1answer
107 views

What makes lattice-based cryptography quantum-resistant?

As opposed to RSA or elliptic curve cryptography?
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Complexity of number field sieve theorem does not match with security of elliptic curves

Number field sieve algorithm can is used to break discrete logarithm on field $F_{p^n}$. The algorithm has time complexity $\exp((c+o(1))\cdot(\log p^n)^{1/3}\cdot(\log \log p^n)^{2/3}$. Originally ...
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Invalid curve attack: finding low order points

Background Here's a description of page 182 of "Guide to Elliptic Curve Cryptography" by Hankerson, Menezes and Vanstone. Here's a quote from that page: The main observation in invalid-curve ...
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1answer
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Complete Set of Test-Vectors for ECDSA secp256k1

Although there are several implementations of ECDSA secp256k1 public available over the internet (the most popular being OpenSSL), it seems that there are no complete set of test-vectors available. ...
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Are there any Secp256k1 ECDSA test examples available?

Are there any available test cases for testing elliptic curves like secp256k1 (Korblitz curves from http://www.secg.org/collateral/sec2_final.pdf)? For curves like P192 there are for example those ...
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1answer
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Given a point $c$ in a field $Z_p$. Can we get another value $c^{'}$ such that $\left(c^{\prime}-c\right)$ is invertible in $Z_p$?

If we have a point in a field $c$. Can we get another value $c^{'}$ such that $\left(c^{\prime}-c\right)$ is invertible in $Z_p$ ?
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1answer
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Understanding elliptic curve point addition over a finite field

I am new to elliptic curve cryptography as well as finite field theory. I am trying to understand point addition in affine coordinates. I understand, that for an elliptic curve $ y^{2}=x^{3}+ax+b $ ...
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Key Value Based Key Derivation

For a system that is using public key cryptography to authenticate users and their actions I'm trying to solve (ease) UI/UX problem so that users will be able to use login/passwords they're accustomed ...
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2answers
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SIDH cryptosystem question

I'm trying to understand the SIDH cryptosystem and got confused at this point: Alice fixes base $\{P_A,Q_A\}$ so that it generates $E_0[l_A^{e_A}]$. Then she chooses secret parameters $m_A,n_A$ and ...
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1answer
59 views

Naming convention for NIST elliptic curves in OPENSSL

NIST standardized 5 elliptic curves (P-192, P-224, P-256, P-384, P-521) for prime fields. When I looked into openssl, these curves are named as prime192v1, secp224r1, prime256v1, secp384r1, secp521r1. ...
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1answer
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NIST elliptic curves behaving anamolous in OPENSSL benchmark

I tried to collect some benchmarks on NIST elliptic curves using charm library. The charm library is just a wrapper over OPENSSL. I experimented with prime192v1 (P-192), secp224r1 (P-224), prime256v1 (...
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1answer
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Is Curve P-384 equal to secp384r1?

I am a bit confused with different notations of elliptic curves. Specifically, I am comparing the NIST specification with the SECG specification. More specifically I want to know if the NIST curve $...
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2answers
735 views

Comparing elliptic curves over prime fields against EC over binary fields

In which scenarios we go for prime fields or binary fields? Please indicate why we would choose one over the other.
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0answers
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Security of an Elliptic Curve Public Key with a “Small” x-coordinate

Consider an elliptic curve over a finite field $F_p$ with $p$ prime and order $n$. Let $Q$ be a generator for the field. Given a public key point $P = aQ$, suppose we have an algorithm that finds an ...
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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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1answer
115 views

Do Weil, Tate, and Ate pairings exist on all elliptic curves?

I don't know much about the math behind elliptic curves. Do Weil, Tate and Ate pairings exist on all elliptic curves? If the answer is negative, then what pairings do MNT, BN and SS curves have? ...
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1answer
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Is there a concept of embedding degree for non-pairing based elliptic curves?

From this post, I learned the concept of embedding degree. Intuitively, if embedding degree of an elliptic curve $E(F_p)$ is $k$, it means there is a way to transform points in $E(F_p)$ to $F_{p^k}$. ...
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1answer
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What does the number 256 in pairing curve BN256 indicate?

There are many pairing based elliptic curves like MNT curves, BN curves, SS curves etc., When we say BN256 curve, what does the number 256 indicate? Is it some group order or number of bits required ...
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Why is the P-521 elliptic curve not in Suite B if AES-256 is?

In the NSA's document, "The Case for Elliptic Curve Cryptography" (archived), we have ...
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2answers
104 views

EC Key Compression

Using the secp256k1 curve, will the below yield the same result? Generate private key -> compress private key -> generate public key Generate private key -> generate public key -> compress public key ...
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3answers
533 views

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 ...
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2answers
926 views

How to calculate a private key from public key on elliptic curve? [duplicate]

Provided, that you have to multiply a well known G by the private key to obtain the public key, why not keep adding G to itself until you reach the public key. The private key should be the number of ...
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1answer
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Question about using Montgomery form for elliptic curve operations on bls12-381

Since the prime for bls12-381 is not of a form to allow easy modular reduction , is the best approach to use the Montgomery multiplication + reduction algorithm? ...
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1answer
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Elliptic curve over prime field with high order roots of unity

Suppose I have an elliptic curve defined over a prime field $\operatorname{GF}(p)$ where $p$ is a large prime (e.g. 256-bit). Suppose also that $p = kn +1$, where $n$ is a relatively large power of $2$...