Elliptic curves are a mathematical structure. In cryptography, it is common to use the structure $y^2 = x^3 + ax^2 + b$ over a finite field. Questions relating to elliptic curves and derived algorithms should use this tag and might also consider specific tags such as discrete-logarithm and ecdsa.

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Is Curve25519-java secure?

I have only about 2 weeks of cryptography experience mostly in the form of questions on bitcoin.se. Is Curve25519-java up to date with current Curve25519 standards? Is Curve25519 itself secure? ...
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189 views

Curve parameter for hyperelliptic curve cryptography?

RFC5639 defines some curve parameter for Elliptic curve cryptography. Aren't there any curve parameter database for Hyperelliptic curve cryptography? What I can only find was that written in this ...
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Smart Card Basics

I want to implement some of the basic encryption algorithms on smart card, could any body guide me how to program a smart card, which tools (hardware and software) I should have, and if these tools ...
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Elliptic Curve is DH function or PKI?

can we reuse same ECC key on TLS for long terms or it must be used just once? (i mean can we use ECC like RSA?) is there patent free ECC implementation ?
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Double-and-add/Montgomery VS blinding

I'm having a hard time understanding why people use constant-time techniques to counter time-attacks, when blinding seems as good and cheaper to implement. Why do people avoid blinding in ECC?
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ECC considered secure in OpenSSL?

this is my first question, please bear with me if it comes across silly. If I openssl ecparam -list_curves on my OpenSSL version (1.0.1f), it spits out the ...
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Polynomial division hardware implementation

I am beginning the implementation of the polynomial binary division algorithm now as I understood i will be checking the MSB bit if 1 to XOR and shift the sum if 0 i will only shift. What I am not ...
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38 views

Modulo Square Roots [duplicate]

Here's my issue and someone can help me understand it so I can program it correctly. I have a point(X,Y) on an Elliptical Curve E(a,b) where a=-3 and B is a large number that is in hexidecimal from ...
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How represent message in Menezes–Vanstone elliptic curve cryptography

I ask about represent message in Menezes–Vanstone elliptic curve cryptography I now encrypt function as follow $C_1 = (M_1 * K_1) mod\ P$ $C_2=(M2 * k_2) mod\ P$ My question is about how much the ...
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DLOG in $\mathbb{F}_{p^n}^*$?

Assume that we are given an element $g\in \mathbb{F}_{p^n}^*$ and $g$ does not belong to any of the smaller subfields contained in $\mathbb{F}_{p^n}$. If the degree of $g$ is some number $q$, how much ...
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Which mathematical operations does secp256k1 point multiplication use?

To convert a bitcoin private key to a public key, the secp256k1 point multiplication math is used. Could I – theoretically – convert a private key to a public key just using the four arithmetic ...
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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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Type G Bilinear Pairings

I was reading PBC and its implementations for finding pairing parameters. I am particularly interested in implementing a BLS signature scheme with 20-byte (160-bit) signatures ("short signatures"). ...
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ECC cryptography with shorter signature when not needing high security?

I am new here and fairly new to cryptography, so if I say something wrong, let me know. I am trying to set up a system where a user can receive a temporary license key over the phone, put it into ...
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103 views

Hardware Implementation of Pairing over BN curves

I am in the middle of FPGA based Hardware architecture design for the computation of Pairing (particularly R-ate Pairing) over BN curves. Where, the point addition, and point doubling should be ...
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102 views

A question about elliptic curves and finite fields in bilinear pairings

Based on what mentioned in the paper "Pairings For Cryptographers" http://www.sciencedirect.com/science/article/pii/S0166218X08000449 the two inputs of a pairing map are two members of two additive ...
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Bilinear pairing

I am working on Efficient Construction of Pairings which are being realized by Miller's algorithm. In this algorithm the basic steps are point doubling and line function computation point addition ...
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Want to use ECC but am clueless [closed]

First off, I'm not an experienced cryptography or computer person, please bear with me. I have some basic experiences with PGP software though (not much of a redemption huh?). I have some data that ...
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Smart card Strong authentication / Verification ( fingerprints)

I'm trying to make a strong authentication software and embedded software in a java card. I have found many papers and publications about the subject… too much information to process and I'm working ...
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141 views

Weil pairing implementation - low level programming language

I'm just started to studying elliptic curve cryptosystems. My one month-goal is to write a simple signature system based on the Weil-pairing. Some parts of it can be written without deeper ...
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74 views

Sextic twist of BN pairing parameters vs security

I've previously asked questions on BN pairing parameters. Here's one more. In the BN construction, one is working in a subgroup of a curve over an extension field $\mathbf{F}_{p^{12}}$ for some ...
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133 views

Safe and computationally efficient way to verify a curve25519 identity?

A client identifies itself as a curve25519 public key. The server wants to verify the client owns the associated private key. Is there a safe and computationally efficient way of doing so? Which ...
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inverse of scalar multiplier in ECC

I am learning to use ECC. i got into situation where i have $Q=abG$, where $G$ is the generator of the finite field formed on EC using a prime $p$ modulus and $a$ , $b$ are random numbers. now suppose ...
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Where can I find examples of ECC implemented in VHDL?

I am looking into implementing ECDSA signature and ECDH key agreement in a Xilinx FPGA. All the examples I have found of VHDL implementations skip over how to construct the low level ECC primitives ...
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102 views

$\phi$ function in Dual_EC_DRBG

I am trying to understand the operation of the Dual_EC_DRBG. I'm reading the formal specification (SP 800-90) and can't seem to find a definition of the $\phi$ function used throughout the definition ...
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About Elliptic Curve ElGamal, 3 simple problems I have trouble with

In Elliptic Curve ElGamal, why are a=b=1 always legal for primes whose lengths are no shorter than 11(2) bits long? Is there any reason why the Point at Infinity can always be encoded as (0,0)? ...
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Does BouncyCastle (for ECC) resist timing attacks?

I need to extend the TLS protocol to be able to use other key exchange scheme based on elliptic curves. I am planning to use BouncyCastle's implementation in Java and in .NET. I am worring about ...
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139 views

Complex Numbers on Elliptic Curves & Usage in Tate Pairing

I'm working with understanding the internals of the Tate Pairing. I was going through an example of the curve $E: y^2 = x^3 + 3x$ over $\mathbb{F_{11}}$. The author is showing the computation of ...
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calculating beta for elgamal elliptic curves [duplicate]

Suppose we use elgamal elliptic curves for secure communication. Bob selects a prime $p$, an elliptic curve $E$, a point $\alpha$ on $E \pmod p$, and a secret integer $f$. Suppose that Bob has ...
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How fast can a wrong decryption key be detected using ECC?

When can a decryption function detect that the ECC key I use for decryption is incorrect? Is it possible to do that during initialization, or does the complete message have to be decrypted to do that? ...
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218 views

Use curve25519 for ElGamal crypto

DJB described curve25519 in his paper which can be found here: http://cr.yp.to/ecdh/curve25519-20060209.pdf. It seems that the main purpose was for Diffie-Hellman key exchange. I think this means that ...
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Diffie hellman key exchange on elliptic curve over an extension field [closed]

I am attempting to do a final semester project where I implement Diffie-Hellman key exchange on an elliptic curve over an extension field (2^256). Can anybody help me to generate the extension field ...
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Is C25519/Ed25519 “twist secure”?

This recent new curve mentions something that's new to me: twist security. http://safecurves.cr.yp.to/bada55.html Are the existing C25519/Ed25519 curves secure against this form of attack?
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What does variable-base point/scalar multiplication mean in ECC?

I am a bit confuse about the term, variable-base point/scalar multiplication, in Elliptic Curve Cryptography. What I have understood so far. It means that the base or point on EC is variable/unknown. ...
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141 views

Elliptic curve parameters

What's the meaning of 160 bit Curve in Elliptic curve ? or 192 or 224 or 256 and etc. And What is the standard for selecting this number of bit ? why they don't say 100 bit curve?
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65 views

Is elliptic curve point multiplication semantically secure?

Is elliptic curve point multiplication semantically secure? I'd like to know if there are some sets of elliptic curve parameters (e.g. NIST curves) that are proved to be semantically secure or that ...
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How to calculate kinv from the given k value

I am implementing an ECDSA NIST test vectors verification application. The test vectors are taken from http://csrc.nist.gov/groups/STM/cavp/#09. One of the test vectors is given below: ...
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Elliptic curve group over a prime finite field $F_p$

If $p$ is a big prime, and the elliptic curve $E$ is defined over $F_p$ by the equation $y^2=x^3+ax+b$ where $a,b\in F_p$. The point on $E/F_p$ together with the infinite point $\mathcal{O}$ form a ...
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Simplified Example of ECC to use in the classroom

I have come up with the following rudimentary example of how ECC relates to asymmetric keys. Is this a valid explanation of ECC and its relationship to asymmetry? To only be deciphered by the person ...
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123 views

What is a good way to demonstrate elliptic curve cryptography?

For school (high school) I am writing an essay on elliptic curve cryptography. The assignment needs to include a practical part, so I decided to write a Python class for elliptic curves. This class is ...
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201 views

Is this EdDSA modification secure?

I am hoping to employ a signed set membership system which is valid iff each signer's contribution to the set is present. The system should allow for two or more mutually exclusive signed sets to be ...
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101 views

Is an elliptic curve over $\mathbb{F}_p$ order preserving for the points $(x,y) \in \mathbb{Z}_p$?

Let $E\colon y^2=x^3+ax+b$ be an elliptic curve, and consider its realisation over the finite field of prime order $p$: $E(\mathbb{F}_{p})$. Then if $0<a,b$ is the following true? $$\forall ...
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315 views

openSSL ECDH private key size

When you are using a named curve like P-256 in openSSL, is there any standard key size for ECDH private key keys? If you look at the ec_key.c file in the openSSL ...
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287 views

Key space: Dense and sparse

I'm taking a cryptography class and I come across these terms dense and sparse key space allt he time. What do they mean? As far as I can I understand, dense key space means that there are more ...
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273 views

Can you help me understand ECC Cryptography and it's algorithm?

I want to know the basic understanding of ECC algorithm for cryptography. But I am not aware of the algorithm. Can anyone provide me with a basic explanation of the algorithm?
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Is there any patent free EC point compression available? If not, should I GPL/LGPL my code?

I'm currently using the BouncyCastle implementation in uncompressed mode, and considering releasing an optimized version of EC code that uses point compression. It appears the only thing limiting me ...
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How do you derive the lambda and beta values for endomorphism on the secp256k1 curve?

You can see a little background about this on this bitcointalk post by the late Hal Finney. $\beta$ and $\lambda$ are the values on the secp256k1 curve such that: $$\begin{align} \lambda^3 &= 1 ...
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C# implementation of curve25519 to ed25519 conversions [closed]

WRT the selected answer here: Can curve25519 keys be used with ed25519 keys? Is there any c# implementation of this or equivalent? Thanks.
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How can i calculate prime of Elliptic Curve?

In many articles i have found directly the calculation of prime elliptic curve. How can i calculate this prime $p$ ? For example if I consider NIST P-256, $ p = 2^{256}-2^{224}+2^{192}+2^{96}-1$. Why ...