# 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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### 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 $g$ has even or odd order. In other words $\textrm{ord}(g) \bmod 2$ can be computed easily. In some cases where the ...
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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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### 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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### Sextic twist over BN elliptic curves

I am struggling to understand how to perform a sextic twist over a BN elliptic curve. This is what I understood so far: Let's consider a BN elliptic curve: $$E: y^2=x^3+b$$ And let's consider a ...
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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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### Precomputation attacks against ECDH

Diffie-Hellman groups are vulnerable to sieving precomputation attacks. These attacks allow a one-time computation against a given DH modulus that makes it practical to attack all subsequent key ...
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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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### 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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### How can I calculate the security level provided by a supersingular Elliptic Curve?

I want to know what security level is provided by an elliptic curve used in Supersingular isogeny Diffie–Hellman key exchange (SIDH). Is there any mathematical convention to follow or by looking at ...
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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 ...
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### What is the (uncompressed) x,y-representation of a curve point on the P-256 NIST elliptic curve?

I am trying to understand the FIDO U2F Raw Message Format, especially the format in which a user public key should be provided. The documentation says the following: A user public key [65 bytes]. ...
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### 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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### EC non-shared cryptosystems - different group for every party

Efficient Identity Based Parameter Selection for Elliptic Curve Cryptosystems by Arjen K. Lenstra contains a proposal for a non-shared elliptic curve cryptosystem. Every party chooses its own field ...
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### EC-ELGAMAL message mapping

I have been able properly set up an EC-elgamal protocol by using algorithms available in an IP that I have developed. Everything works fine, except for the fact that I haven't been able to completely "...
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### 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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### How to map the points of an elliptic curve cyclic group to $\mathbb{Z}_q$ using a hash function?

Let $E$ be an elliptic curve defined over $\mathrm{GF}(q)$, where $q=p^r$. Let $G$ be a cyclic group of points of $E$. Then how we can map points of $G$ into $\mathbb{Z}_q$ using a hash function.
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### Pairings over elliptic curves on rings

Looking at this presentation, Boneh says that elliptic curves could be defined over $\mathbb{Z}/n\mathbb{Z}$ and not necessarily over a prime field $\mathbb{F}_p$ and hence we could define pairings ...
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### Which safe elliptic curves allow for the fastest scalar multiplication

I'm specifically looking for curves that are safe (ideally on this list: https://safecurves.cr.yp.to/) and which allow for the fastest scalar multiplication operations on arbitrary points. By '...
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### Frey-Rück Attack (FR-Reduction) - Tate Pairing

I am trying to understand the Frey-Rück attack and found different ways of a possible implementation. Since I am not yet very familiar with the Tate-Lichtenbaum pairing and the theory of divisors I ...
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### Prime extension field encoding ASN.1

ASN.1 encoding for elliptic curve cryptography is recommended by Certicom, as explained at a related question, covering curves over prime fields and binary extension fields. I'm looking for known ...
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### How are Zhang-Safavi-Susilo signatures short?

ZSS signatures have been introduced by Fangguo Zhang, Reihaneh Safavi-Naini, and Willy Susilo: An Efficient Signature Scheme from Bilinear Pairings and Its Applications, in proceedings of PKC 2004. ...
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### 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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Till now all we know is Hasse's theorem, which states that $|\#E(p)-(p+1)| \leq 2\sqrt{p}$, where $\#E(p)$ is the total number of points in $E_p(a,b)$. Is there any other theorem which defines ...