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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How does ECDHE_RSA key exchange mechanism work?

Using Wireshark, I found these data exchanged with google.com over TLS: Client Hello possible cipher suites and possible curve types (eg. secp256r1) sent Server Hello cipher suite selected ...
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How can I use SSL/TLS with Perfect Forward Secrecy?

I'm new to the field of cryptography, but I want to make the web a better web by setting up the sites that I host with Perfect Forward Secrecy. I have a list of questions regarding the setup of ...
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How can ECDSA signatures be shortened (to be used as a product key)?

So I made my own serial key generation software, using ECDSA, for use in my own applications and it works great so far! To keep the serial key short enough I use a 128 bit EC curve. My final signature ...
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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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59 views

Parameters for elliptic curve prime192v3

I'm looking all over the internet for prime192v3's parameters. I think I may have found them here, but it doesn't say what variable each number matches to. Is there some central place where I can find ...
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317 views

Safe elliptic curve point addition using projective coordinates: How do I tell if the points are the same?

I am trying to implement elliptic curve point addition in hardware for NIST p256 and p384 curves. I have noticed the following issue with the suggested NIST routines: Consider routine 2.2.7 of ...
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49 views

Koblitz encoding a message to a point, what is the “associated auxiliary base parameter”?

I am looking at the Koblitz method for encoding a message as an elliptic curve point. The first step given in the paper I'm reading is: "Choose an elliptic curve and its associated auxiliary base ...
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206 views

Creating serial key generator using ECDSA, how to get signature short enough?

I've written some questions in this stackoverflow and got great responses but now I'm trying to wrap it all together. I have for the last couple of weeks been building a serial key generator project ...
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Are there any asymmetric composite order group bilinear pairings?

Are there any asymmetric composite order group bilinear pairings? Is there a drawback of asymmetric over symmetric bilinear pairings of composite order either in efficiency or in security ?
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43 views

What aspect of elliptic curve encryption paradigms makes them especially susceptible to quantum based attack algorithms?

This was a statement made during a talk at today's DFN-CERT conference but unfortunately it wasn't explained further. Can anyone shed light on why elliptic curves are susceptible to quantum based ...
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44 views

In a additive group is it hard to calculate $bg$ given $ag, g, abg$

The ECDH problem defined that given $g,ag,bg$ it is difficult to calculate $abg$. But it is also difficult to calculate $bg$ given $ag,g,abg$. where $g$ is generator and a,b are elements of group.
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ElGamal with elliptic curves

I've searched some information on ECC, but so far I have only found Diffie-Hellman key-exchange implementations using ECC, but I don't want to exchange keys, I want to encrypt & decrypt data like ...
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83 views

What are the computational benefits of primes close to the power of 2?

Recently I was reading some article about the Bernstein's Curve25519. This is a particular Montgomery curve over $\mathbb{F}_q$ where $q = {2^{255}-19}$. What I missed or was unable to understand is ...
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162 views

Is cryptanalysis of CTB-Locker really impossible?

It seems that CTB-Locker make a lot of victims nowadays, and yet, the full encryption scheme of it is now publicly known [1,2]. Would any of you could find a weakness to exploit in this encryption ...
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108 views

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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Explanation of each of the parameters used in ECC

I'm having a very difficult time finding a clear explanation of the parameters used elliptic curve cryptography. I know for certain that $p$ is the number or order or whatever of the given field that ...
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91 views

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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Why does NaCL have different keys for signing and encryption?

I want to start using NaCL to sign messages that will go into a message queue, and I noticed that it generates different keys for each operation. Is there a reason for this? Can I not use the same PK ...
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Given $g^a, g^b, g^c, g^{1/b}$, is it hard to distinguish $e(g, g)^{abc}$ from a random value?

where $g$ is a group element in bilinear group $\mathbb{G}$. I understand it is very similar to the conventional DBDH problem, but $g^{1/b}$ is also known, possibly making it easier? Does anyone know ...
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In ECC, how do I prove that point addition is commutative?

I am studying elliptic curve cryptography and this question is related to the commutative property of point addition operation. Point addition $P_3(x_3,y_3)$ of two points $P_1(x_1, y_1)$ and ...
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135 views

Why are some $x$ coordinates unsuitable for an ECDSA generator point?

For Bitcoin's ECDSA curve (secp256k1, where $a=0$, $b=7$), why can't the generator point's first coordinate be $x=0$? That is, the point on the curve would be $(0,y)$ where $y$ satisfies $y^2 = 0^3 + ...
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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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Elliptic Curve based blind signature implementation

I want to use Elliptic Curve based blind signature scheme for my research. There is no proper implementation of ECC-based blind signatures. Can someone describe to me which things I need to follow ...
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Is there a flaw in this ECC blind signature scheme?

Recently I've found the following work on the internet: An ECC-Based Blind Signature Scheme The paper claims to be an ECDSA blind signature however it seems that their scheme has a flaw in it. The ...
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Point decompression on an elliptic curve

I'm programming an elliptic curve cryptosystem and I'm having difficulty with decompressing points. The following information is from my project specification as to my understanding: Given a point ...
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135 views

Isn't the security of EC curve 25519 126 bits?

The security of the EC25519 is given as 128 bits, but since the order of the group is 252 bits shouldn't the security be 126 bits? Given as half the magnitude of the underlying field, since DLP ...
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curve25519 weak points for contributory behaviour

The Diffie-Hellman on curve25519 is usually calculated using the base point $(9,…)$ which induces a cyclic subgroup of $G:=\{\infty\}\cup(E(F_{p^2})\cap(F_p\times F_p))$ with index 8, i.e. there is a ...
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114 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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315 views

How does ECDH arrive on a shared secret?

I read a brilliant, three part article on Elliptic Curve cryptography (one, two, three). It was able to explain Elliptic Curves to me in a way that didn't require a math degree to understand. The ...
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324 views

How do I convert the definition of E-521 into a curve definition a la Bouncy Castle?

I am currently trying to create an ECCCurve for E-521. Unfortunately, it is not currently a named curve in the library I am using, so I will have to define it manually. I am using the definition of ...
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178 views

Is this ECC encryption key sharing method okay?

Is this encryption key sharing okay to use? Or is much better to use ECIES? $G$ = base point $a$ = Alice’s private key $b$ = Bob’s private key $A = aG$ = Alice’s public key $B = bG$ = Bob’s public ...
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Are there any Secp256k1 ECDSA test examples available?

Are there any available test cases for testing elliptic curves like secp256k1 (Korblitz curves from http://www.secg.org/collateral/sec2_final.pdf)? For curves like P192 there are for example those ...
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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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Solving Quadratic equations in Galois Field (2^163)

Hello I am working on implementing a message to elliptic curve point mapping hardware circuit I have done some research and found out the koblitz mapping method: I will be using a field of binary ...
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Which multiplicatively homomorphic encryption scheme supports encryption of 0?

I want a multiplicatively homomorphic encryption scheme that supports encryption of 0 (e.g. Elgamal doesn't support). I also want the multiplication to be operated on the ciphertext of 0, i.e., if ...
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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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171 views

Can keys from Bitcoin's Hierarchical Deterministic Wallets be correlated (reducing privacy)?

I'm trying to understand if the feature "Hierarchical Deterministic Wallets" in Bitcoin allows for complete privacy of all derived keys, and if any of those keys can be associated with each other ...
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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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request for data to test deterministic ecdsa signature algorithm for secp256k1

I’m implementing the RFC 6979 procedure to compute a message signature. I want to test my program on the secp256k1 elliptic curve. Note the “k” in secp256k1, i.e. the Koblitz curve. If you have the ...
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processing time for multiplication and exponentiation in pairing base cryptography

I'm using the Boneh-Boyen-Shacham signature scheme and want to estimate complexity in my scheme. As reported in "Scott M., Efficient Implementation of Cryptographic pairings", if we set parameters ...
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Polynomial Inversion over Galois Field

Hello guys I am looking to calculate the Inverse of a given polynomial in Galois field I have found the little Fermat's algorithm and the Itoh-Tsujii I am getting a bit confused with both algorithm ...
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Roots in modulo field

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 hexadecimal from -51BD. To compress this point oficially in a program, we know that every $X$ ...
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35 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 can I change representation of point to Jacobian coordinates in Edward's Curve

I want to simulate this algorithm but I want to change it's output to Jacobian coordinates. what should I do ? In the other way how can we change extended homogeneous coordinates to Jacobian ...
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Find generator for irreducible polynomial over binary field

I read this tutorial and I have following question. How they assume that generator: g = (0010) is correct for this polynomial and how to choose the best generator from all for the field.
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ECC - ElGamal with Montgomery or Edwards type curves (curve25519, ed25519) - possible?

I know the usual way of using getting shared secrets for encryption with ECC is DH, however, this only works with two keypairs of exactly the same kind, for example two curve25519- or two ...
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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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Finding Elliptical curve points and encoding text using them

I recently got into learning Elliptical curve cryptography and are currently building a project in C#. Everything is working well so far, I can encode and decode points, and thanks to this forum I ...
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126 views

Elliptical curve cryptography key generation time

I am currently trying to learn more about Elliptical curve crypthography and have finally started to get things working and undestanding the different pieces. I've written a small project in C# and ...
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What is the difference between order of base point and curve order in EC? [duplicate]

When I was read about the elliptic curve cryptography I found some definition about domain parameter of elliptic curve like the follow. But I did not understand something $p$: prime number. $a, b$: ...