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Questions tagged [elliptic-curve-generation]

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45 views

Difference in elliptic curve order and finite field size [duplicate]

Must the prime finite field, Fp, an elliptic curve is defined over always have a greater number of elements than the cardinality of an elliptic curve. For example, If I have ...
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1answer
70 views

Point doubling with only one coordinate

In many source codes that implement ECDH, there is a function that multiplies the base point of that curve with a constant. This function usually takes as arguments the constant and just one ...
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79 views

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
486 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
41 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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53 views

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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28 views

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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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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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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61 views

ECC - complex multiplication and key agreement

I'd like to ask three questions - 2 of them regard CM method. The last is regarding the ECC domain parameters generation on the fly, see https://eprint.iacr.org/2015/647.pdf What role has ...
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2answers
125 views

How to model a miniature elliptic curve?

For educational purposes I would need to work on an elliptic curve that has a small field, but holds the safety futures of a real curve. Is that possible to have such a curve!? For example for the ...
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170 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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100 views

NIST ECC Curves without pairings

NIST FIPS.186-4 has standardized 5 ECC curves on field $\mathbb{F}_p$ (P-192, P-224, P-256, P-384, P-521) and 10 elliptic curves on binary fields. I need to use ECC curves without pairings for my ...
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1answer
187 views

What math should I learn to get in depth with Elliptic Curve Cryptography research?

My background is computer scientist. I have done applied cryptography research for a while. Currently, I'm working on Elliptic curve cryptography. To understand the idea and how to use Elliptic curve ...
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1answer
265 views

C# equivalent of openssl ecparam -name prime256v1 -genkey -noout -out [closed]

I am having some issues generating a proper private key,I need it to be an Elliptic Curve private key suitable for use with NIST P-256 which i than need to convert to Base64-encoded private key in ...
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1answer
710 views

How were the P-256 parameters chosen?

In NIST FIPS 186-4 (page 90), it is said that $c$ is the output of SHA-1 on a seed that was chosen randomly. Then the parameter $b$ of the EC is chosen, according to the formula: $$b^2 \times c \equiv ...
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1answer
380 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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2answers
148 views

What are the properties of secure Elliptic Curves?

I have heard about the standard elliptic curves called NIST curves. What are the properties of such cryptographically secure elliptic curves? Are they standardized according to certain protocols? Also,...
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142 views

Cryptographically Secure Elliptic Curve

What are the properties a cryptographically secure Elliptic Curve must have? I have started to create a list and wanted to know if I forgot some important points, and if it is correct so far: A curve ...
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2answers
2k views

How do malware rely safely on ECDH algorithm to maintain secrecy of keys?

In traditional malware (especially ransomware) using RSA approach, the public key may be hard-coded in the malware binary and is used to encrypt a symmetric key generated on the system. The symmetric ...
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1answer
1k views

What is the curve type of SECP256K1?

This is possibly a dumb question. I'm trying to input SECP256K1 curve parameters to a system that expects any custom curve. The form is asking for "curve type". It offers three options: Short ...
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1answer
808 views

Curve point generators for ECC

I'm working on a cryptosystem based on ECC (elliptic curve cryptography). I need to choose several (~10) curve points which should be used as generators, and it's important that their relation is ...
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1answer
279 views

OpenSSL - creating random elliptic curve and measuring performance

I want to generate random elliptic curves (over $\mathbb F_p$ and $\mathbb F_{2^m}$) using OpenSSL, that's my task. At first curves over $\mathbb F_p$. Now I am able to generate prime $p$, $a$ and $b$...
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1answer
190 views

How is the precomputed table for 25519 Elliptic curve generated?

I am wondering how the precomputed table for scalar multiplication for elliptic curve (in my case 25519) is generated/precomputed? I am talking about this [https://github.com/WhisperSystems/...
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1answer
107 views

finding a from a*G

This might be a newbie question, In elliptic curve, I chosen a secret a and computed aG, where G is the base point. But here since I know G, I can compute inverse of G and multiply it with aG, to get ...
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804 views

What's wrong with this curve (generation algorithm)?

In this tweet, Paulo Barreto proposes the following elliptic curve over $\mathbb{F}_{2^{255}-19}$: $$ E_\mathrm{PB} : y^2 = x^3 - 3x + 13318 $$ with $G_\mathrm{PB} = (-7, 114)$. Now I would like to ...
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1answer
2k views

What is the relationship between p (prime), n (order) and h (cofactor) of an elliptic curve?

I am reading up on ECC and having trouble understanding how these are related. In a finite field, all point operations are taken modulo $p$. $n$ is the order of the generator $G$ — which apparently ...
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2answers
309 views

Finding if two points on elliptic curve are related

On a given elliptic curve I have some points that are defined like this: Where $$A=a*G$$ $$B=b*G$$ $$C=(a+b)*G$$ $$D=d*G$$ $$E=(a+d)*G$$ So finally I have two equations like below: $$A+B=C$$ $$A+D=E$$ ...
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1answer
341 views

Bilinear Form: Weil pairing

I am doing an example on Weil pairings, and for that purpose I follow the thesis of Alex Edward Aftuck, The Weil Pairing on Elliptic Curves and Its Cryptographic Applications. By following his thesis,...
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1answer
573 views

How to check whether the given curve is Elliptic or not?

I've given a curve E5(2,3) over Zp. How to check this curve is Elliptic or not? I know the curve formed by this will be: (y2) mod 5 = (x3 + 2*x+3)mod 5.
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316 views

Why are there so many different elliptic curves?

There are Chinese, French and NIST curves. There's a Million Dollar one. The BADA55 Research Team studied 1 million variants. Some are based on widely different formulae. Indeed there are entire ...
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1answer
114 views

Order of the curve and generator

Does the order of the curve and the order of generator should be coprime for an elliptic curve defined over a prime field?
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2answers
2k views

Security of elliptic curves

How can we say an elliptic curve is secure and can be used for cryptographic applications?
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1answer
153 views

Order of an elliptic curve defined over a prime field

I found the following algorithm to find the generator of an elliptic curve: Find the order of the curve - N. Choose any random point on the curve - P. Find the order of that point - n. Calculate co-...
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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 ...
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2answers
692 views

How can I find the generator of a composite group and $Z_p*$?

I was doing some research on elliptic curves. I know how to find the generator of $Z_p$ (this is a prime group). But I came across the term $Z_p*$ (group containing elements that relatively prime to $...
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120 views

Does the generator (base point) is preshared between the sender and receiver in elliptic curve cryptography?

Or the elliptic curve and the generator point are known to everyone?
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2answers
479 views

Is the prime P is fixed for an elliptic curve defined over a particular prime field F_p?

I have seen the NIST, SEC and Brainpool standards. They have used same prime for a particular bit curve (128,192,256,521). Is the prime value fixed for a particular security (field size)?
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2answers
821 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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1answer
4k views

How to find the generator of an elliptic curve? [duplicate]

If the elliptic curve has prime order of points, then all of its points are generator. Is this true? If so, how can I find the optimized generator(which generates more number of points) among them?
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1answer
421 views

Supersingular vs non-singular (Same or different)

I have a frustration for the term "supersingular" elliptic curve and "non-singular" elliptic curve. In some paper, performance evaluation is done using supersingular elliptic curve and in some cases, ...
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1answer
511 views

Security level difference: supersingular vs non-singular elliptic curve

In some evaluation of elliptic curve cryptography, it says that for same security level In supersingular curve over $F_p$ with group of prime order $q$, p=512, q=160 bits In non-singular curve , p=...
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1answer
270 views

Elliptic curves with field sizes that not byte-aligned

Why there are abnormal field size like 521, 571, 233, 283 bits in prime and binary fields that are defined by NIST?
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2answers
324 views

Elliptic curve with non-prime generator?

Suppose we need a special Elliptic curve on prime field $Fp$, for some reason, that the order of the generator must be $6*q$, where q could be a big prime, does this weak the ECC encryption? What's ...
3
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1answer
506 views

Short Weierstrass equation is non-singular for not 2 or 3 characteristic

Consider a field $K$ of characteristic $p \neq 2,3$. Consider a curve $E$ over $K$ defined by the equation $y^2 = x^3 + ax + b$. How can I show that: $E$ is not an elliptic curve (it is not ...
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0answers
240 views

TypeA pairing, elliptic curves in pairing based cryptography

I am beginner to pairing-based cryptography. After downloading jpbc library, curve parameters files are seen as properties file. For example, for type A curve, following parameters are given. type a ...
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2answers
96 views

Table of Curve Paramters

I'm studying Elliptic Cruve Cryptograhpy. When I do a search on Google of ECC, I find some pdf where I see these curve's paramters: $q, h, r, exp1, exp2$. What are these parameters ? Are there tables (...
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3answers
378 views

What's up with unnamed elliptic curves in e-passports?

At my work I deal with the cryptographic aspects of the international E-Passport specification (the crypto chips embedded in your passports, the kiosks at airports that talk to them, and the ...
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2answers
185 views

Is there a theorem to determine the elliptic curve parameters based on the group order?

By Hasse's theorem we know that range of the group order of the elliptic curve. And similarly, there exist a theorem on the admissible order of elliptic curves. Suppose by the theorem on the ...
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1answer
309 views

What is necessary for generating an elliptic curve?

Let's say I want to generate my own elliptic curve with an order whose bit length is $n$ (specifically 2048, 4096, and/or 8192)? How would I do this? What needs to be done? What software can do this? ...