# 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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### Software timing attack using Kocher method

What's the minimum number of random sample points needed in Kocher's timing attack, so that we can determine enough valid measurements of $A_{i,r}$ and $D_{i,r}$? I'm working from this paper: Volker ...
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### About a SETUP mechanism on ECDH

I'm following these three articles: Kleptography: Using Cryptography Against Cryptography, Kleptographic Attack on Elliptic Curve Based Cryptographic Protocols and Elliptic Curve Kleptography . In ...
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### Which is the smallest safe elliptic curve (bit-length)?

At https://safecurves.cr.yp.to/ some elliptic curves are listed which passed certain security tests. The smallest bit-length of a safe curve listed there is 221 bits. At wiki page discrete logarithm ...
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### Parity of the order of a element

Given an element $g$ in a cyclic group $G$ of known order $m$ its easy to test if $m$ has even or odd order. In other words $\textrm{ord}(g) \pmod 2$ can be computed easily. In some cases where the ...
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### 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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### Regarding the need to hash the shared secret in X25519 with the public keys

I was looking at the LibSodium documentation where it says [...] and to mitigate subtle attacks due to the fact many $(p, n)$ [public key - secret scalar] pairs produce the same result, using the ...
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### 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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### Fastest known Elliptic Curve Cryptography “solution” (coordinate systems (multiple?), algorithms, precomputed values etc)?

I am writing an Elliptic Curve Cryptography SDK in pure Swift, and currently I am only using Affine Point and simple Double-and-add. I am soon about to work on a faster solution. I am asking for help ...
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### Encrypt using ECDH with two different EC public keys, minimizing payload size

Let's say Alice has the private EC keys $a$ and $b$, with a base point of prime order $G$. Alice computes the corresponding public keys $A = aG$ and $B = bG$, and sends them to Bob. Bob now wants to ...
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### Elliptic Curve Blind Signature Implementation

I have seen this prior post: Elliptic Curve based blind signature implementation Currently I'm sizing up how difficult it would be to attain Elliptic Curve Blind signatures for an application I'm ...
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### As a cryptographer, what are the things I should care about in my implementation of pairing functions?

As a beginner in cryptography, I do not know anything about different pairing types more than their names. So far, I know these names: Ate pairing, tate pairing, eta pairing, and r-ate pairing. I am ...
170 views

### ECIES: Purpose of optional shared information?

According to Wikipedia the ECIES algorithm has two optional shared information $S_1$ and $S_2$. They are used as follows: Generate a random shared secret $Z$ according to ECIES, which will never be ...
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### Using the same private key for two ECC key pairs

Let $(d_1,Q_1)$ and $(d_2,Q_2)$ be ECC key pairs over two different elliptic curves (say NIST P-224 and NIST P-256). According to the Elliptic Curve Discrete Logarithm Problem (ECDLP), if the private ...
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### Attack on Weierstrass Elliptic Curve

I have a naive question(as non specialist in this field). While reading Weierstrass Curve description,I found that it turns into 2 periodic tori on 2D complex plane. Is is it possible to create ...
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### Security of ECC over finite fields of characteristic $p\approx2^{50\pm10}$?

What's the security of Elliptic Curve Cryptography over finite fields of word-sized characteristic $p\approx2^{50\pm10}$? We are talking about $\Bbb F_q$ where $q=p^k$ for some suitable $k$. ...
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### Is this distributed random oracle scheme safe?

This question comes from an issue raised in another question: Non interactive threshold signature without bilinear pairing (is it possible)? Is the proposed random oracle model safe when trying to ...
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### ECC: Lightweight proof of correct exponentiation

In the context of ECC. There's an EC point $P$ which is supposed to be a known power of another known point $G$ (generator). That is: $P = [k]G$ (in additive notation) This should be verified on an ...
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### Using (EC)DH to generate a signature

Say I have access to a system A that is limited to performing (EC)DH, followed by key derivation to produce a secret key. This secret key is later used to provide integrity protection. There is a ...
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### What are some use cases for white-box digital signatures?

There were 2 papers published in the last year, that describe 2 different white-box identity-based digital signature schemes: White-Box Implementation of the Identity-Based Signature Scheme in the ...
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### How to use Montgomery arithmetic for elliptic curves (FIAT cryptography)

Let us consider the source code for curve P-256 from BoringSSL. This source code can be found here. This source code uses the FIAT generated implementation for field arithmetic. This implementation ...
109 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 PlĆ»t it is described how you generate valid parameters for the SIDH ...
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### If curve bn256/bls12 support the isomorphism from $G_2$ to $G_1$?

Is bn256 or bls12 a type-2 pairing-friendly curve? As Dan Boneh said here While in many pairing instantiations this Ļ exists naturally, in some instantiations it does not. However I can not find ...
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### How hard is it to link a hardened BIP32 child public key to it's parent public key?

The BIP32 spec says this in the security section: ...
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### Elliptic curve of order $p = 2q + 1$

Does anyone know an example of an Elliptic Curve of caracteristic $p$ ($E_p$) that has a point generator $G$ that generates a subgroup of order $q$, with $p$, $q$ being prime numbers and $p = 2q + 1$?
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### About the scalar multiplication on Koblitz curve in FIPS PUB 186-4 (2013)

In FIPS PUB 186-4, the computation of scalar multiplication on Koblitz curves is given in p.106~109. In p.109, step 11.3, $(r_0,r_1)$ is updated with $(r_1+\mu\,r_0/2,-r_0/2)$. But under ...
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 ...