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Questions tagged [elliptic-curve-generation]

2
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2answers
96 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 ...
3
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0answers
54 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 ...
0
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0answers
44 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 ...
0
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1answer
130 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 ...
0
votes
1answer
95 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 ...
1
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1answer
167 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 ...
4
votes
1answer
138 views

Generating new Koblitz curves

is there the way how to generate new Koblitz curves (over F2n and Fp as well)? I found that "The recommended parameters associated with a Koblitz curve were chosen by repeatedly selecting parameters ...
1
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2answers
108 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,...
3
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0answers
103 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 ...
10
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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 ...
7
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1answer
990 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 ...
3
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1answer
337 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 ...
1
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1answer
200 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 $b$...
0
votes
1answer
134 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/...
1
vote
1answer
96 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 ...
6
votes
1answer
789 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 ...
7
votes
1answer
636 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 ...
0
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2answers
159 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: $$A+B=C$$ $$A+D=E$$ ...
1
vote
1answer
207 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,...
0
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0answers
97 views

Security of Elliptic Curve Cryptography using supersingular curve [duplicate]

If elliptic curve cryptography is implemented using supersingular curves, the security is weaker than implementation using non-supersingular curves. I don't know why. For the same security level, it ...
0
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0answers
73 views

Performance difference: supersingular vs non-singular elliptic curve

In some evaluation of elliptic curve cryptography, https://link.springer.com/article/10.1007/s00500-016-2231-x, it says that for same security level In supersingular curve over $F_p$ with group of ...
1
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1answer
350 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.
7
votes
1answer
274 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
79 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?
2
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2answers
1k views

Security of elliptic curves

How can we say an elliptic curve is secure and can be used for cryptographic applications?
0
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1answer
116 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-...
4
votes
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 ...
2
votes
2answers
439 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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2answers
105 views
3
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2answers
402 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)?
1
vote
1answer
527 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.
2
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1answer
3k 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?
0
votes
1answer
321 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, ...
4
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1answer
368 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=...
5
votes
1answer
214 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?
3
votes
2answers
291 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
votes
1answer
387 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 ...
3
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0answers
195 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 ...
-1
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2answers
92 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 (...
10
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3answers
341 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 ...
6
votes
2answers
157 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 ...
1
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1answer
224 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? ...
2
votes
1answer
87 views

How to convert roots of Weber polynomial to Hilbert class polynomial over modulo prime?

Using any non square root discriminant $D$, we should be able to find the Weber class polynomial. How can I convert the roots of a Weber polynomial to a Hilbert class polynomial over modulo prime?
3
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1answer
301 views

Counting points on elliptic curve over binary field

How to count number of rational points on elliptic curve over binary field?
7
votes
1answer
2k views

Curve25519 vs “Million Dollar Curve”

Quoting from the Million Dollar Curve website: By using publicly verifiable randomness produced in February 2016 by many national lotteries from all around the world, we propose to generate a ...
4
votes
2answers
658 views

Implementation of ECC over binary field

I am supposed to implement ECC over binary field (in C++) for equations of the type - $y^2 + xy = x^3 + ax + b$, as my project. I wish to include the following features : The user will enter a prime ...
4
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1answer
2k views

Do Weak Elliptic Curves Exist?

I'm running into very contradictory opinions when try to understand if weak elliptic curves exist. I'm not interested in the case when a curve's weakness is attributed to properties of an EC's prime, ...
6
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3answers
506 views

Safe curves in Weierstrass form?

I would like to implement a protocol using elliptic curves. I'm thinking of using MIRACL so using curves in their Weierstrass form is preferable as it they are supported by this framework. I don't ...
17
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1answer
2k views

How to calculate elliptic curve parameters?

I'm having a rough time understanding the math behind elliptic curves. I want to implement ECDH where user can define a, b, and p parameters of elliptic curve. How can I calculate generator base ...
13
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1answer
916 views

How to generate own secure elliptic curves?

I know that the algorithm used to generate the Brainpool curves and the NIST curves is published. The algorithm should be this one (RFC5639 Appendix A). From what it looks like it's rather slow to ...