# Questions tagged [elliptic-curve-generation]

ECC is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys compared to non-EC cryptography (based on plain Galois fields) to provide equivalent security. Elliptic curves are applicable for key agreement, digital signatures, pseudorandom generators, and other tasks. they can be used for encryption by combining the key agreement with an asymmetric encryption scheme.

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### What does AZ^4 = -3 (mod p) mean in the context of elliptic curve cryptography?

In the documentation of TeraFire® cryptocore, the following statement is given (page 14): For For elliptic curves of the form y² = x³ + Ax + B, if there exists a solution to AZ^4 = -3 (mod p), then [....
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### Using a public key's points (other than the generator point) to calculate the order of the group (SECP256k1)?

Imagine if we were on a mission to try to calculate the order of the cyclic group $n$ ...
1 vote
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### in elliptic curve over finite field we get two values of y for each x now how can we draw elliptic curve using these points?

as we can see in the image one side elliptic is wrapping around torus in clockwise and other in anticlockwise direction my question is when we find two values of y for each x how can we know which y ...
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### Getting the slope of a public key given its x and y coordinates

Is it possible to get the slope of a public key given its $x$ and $y$ coordinates? Since all the ECC calculations come from geometry, I thought this calculation might be possible.
1 vote
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### Example of elliptic curves endomorphism construction

I've started learning about complex multiplication (CM) on elliptic curves. For clarity (and intuition), I want to make some basic example of elliptic curves endomorphism construction for a concrete ...
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### Is it possible to generate an elliptic curve (with the hard discrete logarithm problem) by iterating only a finite field, but not its $j$-invariant?

Let me ask one question. Maybe, you know an answer. Thanks in advance for any response. Let's fix an elliptic curve $E$ over the field $\mathbb{Q}$ of rationals without complex multiplication, i.e., ...
1 vote
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### Creating a weak BLS

I am exploring the implementation of a cryptographic signature that users can manually input, and I am willing to compromise security to a level where a hacker investing several thousand dollars in ...
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### Generating pseudorandom numbers using Dual_EC_DRBG

I am currently learning about the Dual_EC_DRBG protocol and I am stuck at the calculation of the initial state with the point P. For context, I am using the secp256k1 curve with a = 0 and b = 7. I ...
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### Standard Montgomery curves over prime field

Is there some source of standard, vetted, efficient Montgomery elliptic curves over prime field? I'm looking for curves $B\,y^2\equiv x^3+A\,x^2+x\pmod p$ engineered for efficient computation of ...
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### Generating a new curve using an existing curve and new prime

Can you take a curve equation from https://safecurves.cr.yp.to and a large safe prime from existing DH parameters (for example openssl dhparam 9000), combine them, ...
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### EC public key with leading zeros

Let us take example of secp256k1 curve. The current known public key with most leading zero (in x cordinate) is: ...
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### Practical deployments of ECC with cofactor of elliptic curves $4$ or $8$?

Are cofactor $4$ and $8$ ECC schemes widely used in practical deployments such as those in cryptocurrencies? Can you name some practical settings where there curves are used and cryptocurrencies where ...
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### Non-interactive EC DKG (Distributed Key Generation) question

Normally, when computing an EC threshold DKG, I have all parties reveal a commitment to the public key, and only reveal their own public key after verifying the commitments. Otherwise it's trivial ...
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### How to generate a random point on an elliptic curve without knowing it's corresponding scalar private key

Given an elliptic curve with generator $G$, is it possible to generate a random point on the curve $Q = a \cdot G$ without knowing the secret value $a$ that generated it? Note that just using an $a$ ...
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### In The Ristretto Group, do all points sampled with Elligator have the same order?

Assume the Hash-to-ristretto255 function Elligator as laid out here. Assume a random hash that is then mapped to a point in the <...
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### Deterministic Key using a seed for ECNamedCurve

I am trying to generate deterministic keys using EC curve secp521r1 in java. I went through KDF but, couldn't find any references to use it with EC curves. I would appreciate if someone could point me ...
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### Break El Gamal for Elliptic Curves

There is an elliptic curve El Gamal digital signature scheme. Alice fixes an elliptic curve $E$, a prime $p$, a point $A$ on $E$, a secret integer $a$, and computes $B = aA$. She makes \$(E, ...