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 ...

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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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309 views

Simple digital signature example that one could compute without a computer?

I am working on a document to explain Bitcoin to students. But I am having a hard time translating the principle described in §2 of the Bitcoin whitepaper in layman's terms. There is a great question ...
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1answer
41 views

Given $g^a, Y$, is it hard to distinguish $e(g,g)^{ab}$ from a random value?

where $g$ is a group element in bilinear group $G$ $Y = M.e(g,g)^{ab}$ $M$ is a message Does anyone know the answer or suggest some material for reference? Many Thanks
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1answer
115 views

Can j-invariants be used to decide which elliptic curves are suiteable for cryptography?

The j-invariants classify the elliptic curves up to isomorphisms (if we suppose to work in the algebraic closure). Is this classification used in some way to decide whether or not an elliptic curve ...
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78 views

How to define order according to domain parameters in elliptic curve pairing groups

According to domain parameters, as an example Type 1 pairing domain parameters are ...
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1answer
316 views

RSA and ECDSA Certificate Sizes

Is there a table (or a whitepaper from official sources) that compares the size of X509 certificates generated with RSA (starting from 1024 bits) and ECDSA (starting from 160 bits) ? Thanks for the ...
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93 views

Why is there no 'ECDSA' version of 'DHE-RSA-CHACHA20-POLY1305'?

So I was just checking my TLS cipherlist and noticed that there was a 'DSS' / DSA / ECDSA version of every ...
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43 views

Clarify EC point addition and multiplication

Please clarify the below doubt regarding EC point addition and multiplication: $P$-Generator Point; $a$ and $b$ are integers; $X$ and $Y$ are EC points, defined as follows: $X = (a*P) + (b*P)$ $Y = ...
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104 views

Is this a secure method of encrypting with authentication?

My goal is to allow two clients to send files securely over an untrusted network without the need for more than one block of information to be sent. Both clients have ECDSA keys of size 256 bits. I'd ...
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60 views

Elliptic curve point addition

I have this curve in E(F131) : $$Y^2 = X^3 + X + 2 $$ I want calculate the sum P + Q considering that $$P= (5,1)$$ and $$ Q = (60, 49)$$ For calculate the result i use these formulas: $$ xr = ((...
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54 views

EC based password authenticated key exchange protocol

I would like to know: What is the currently best (efficiency and security wise) known EC based password authenticated key exchange protocol? (i.e. is there any EC based protocol may be similar to DH ...
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82 views

Is there any pattern in points on EC?

I read some where in crypto.stackexchange answer (related to EC-SRP protocol) that there is pattern in points on elliptic curves, i.e. if given some points containing mix of correct and wrong points ...
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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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201 views

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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0answers
63 views

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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89 views

advantages of hashing over elliptic curve signatures for a proof of work protocol

I'm trying to create a proof-of-work protocol for a proof-of-concept software, and it's basically something like this: ...
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0answers
114 views

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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441 views

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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174 views

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"). So,...
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132 views

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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0answers
141 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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154 views

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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60 views

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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110 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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153 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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232 views

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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234 views

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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112 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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166 views

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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313 views

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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0answers
161 views

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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26 views

elliptic curve and embedding degree

I am new in ECC. I am confused what the embedding degree in elliptic curve represents and what is the impact of its values on the curve and security ( small values or large values? What does the ...
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44 views

Found weil pairing. Index Calculus method on the results of weil pairing

Consider Elliptic curve with p = 59, A = 1, B= 0, P = (25,30) and Q=(35,31). So I tried to solve this using MOV attack. The torsion point for them E[5] is R(-25,30i) where is sqroot -1 Chosen two ...
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38 views

Has anybody implemented BGN cryptosystem to multiply two plain texts?

I have tried to use this code as follows, http://stackoverflow.com/questions/33581962/bgn-implementation-in-java but the issue is, when it generate the pairing, the output is in GTfiniteelement (for ...
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55 views

Weil Pairing - Miller's Algorithm

I'm trying to implement Weil Pairing using Miller's algorithm. I have got couple of questions. How to select $m$? As stated in this link page 13, I interated from $1$ to order of point $P$ such that ...
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59 views

Smart Card choice for PKI implementation

I'm seeking to implement a national digital signature standard on a smart card. I feel like this is a good place to ask if anybody is acquainted with a hardware supplier offering smart cards that ...
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40 views

How common are non-RSA digital certificates?

Is there a statistic available that shows just how common are DSA or ECC certificates amongst webservers? I know that RSA-based certificates are the most common, however I'd like to know, if there is ...
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0answers
75 views

Is there any difference between NIST and SECP curves in-terms of their algorithms and implementation?

I'm implementing ECDSA for NIST P-256 curve. I just want to know the same implementation will work for SECP curves also?. If it doesn't, then please suggest me references of algorithms or sample ...
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57 views

Simulation of a custom build network security algorithm with ElGamal Cryptosystem using Elliptic Curve

I am trying to build an algorithm to encrypt and decrypt text using ElGamal Cryptosystem using Elliptic Curve. My algorithm generation is done. But at simulation part I stuck. My algorithm steps are ...
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26 views

Do most TLS 1.2 implementations express curves in a canonical form when performing EC arithmetic?

Sorry if this is a silly question, but does anyone know if the cryptographic libraries which implement TLS 1.2 for Firefox, Chrome, etc. express a given curve in a canonical form (i.e. one of ...
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95 views

Degrade in performance with SSL_OP_SINGLE_ECDH_USE?

We have used SSL_OP_SINGLE_ECDH_USE when setting up our SSL_CTX . This seems to be causing a degrade in performance. I'm not able to find proper documentation for it except that it generates new ...
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80 views

Semaev summation polynomials

I am little confused how this attack works. We have the points $P, Q$ such that $Q = nP$. We let $u_{1} $and $u_{2}$ such that $R(x,y)=u_{1}P+u_{2}Q$. Then if we find the solution $x_1,...,x_n$ of the ...
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34 views

Using Montgomery ladder to calculate the coordintes

In one of my assignments I need to solve the below: For a Montgomery curve $3v^2 = u^3+u^2+u$ over ${\mathbb{F}}_{11}$ and point $P = (9,8)$. I need to compute $x$ coordinate of $3P$ using ...
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55 views

Translation of Schorr Ring Signature to ECSchnorr Ring Signature

I have to write an EC version of Schnorr Ring Signature Scheme. I've already wrote regular ECSchnorr Signature Scheme using this (page 128). I've found a scheme of Schnorr Ring Signature Scheme (page ...
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How to sign an elliptic curve point using an ECC signature scheme?

In Schnorr based ECC signature scheme, a message $M$ is signed with the private key $\mathit{sk}$ as $$s=\mathit{sk}\cdot h(M,R)+k$$ where $R=k\cdot P$ and $P$ is a base point. If $M$ is a point $(x,y)...
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112 views

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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128 views

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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91 views

Elliptic curve trapdoor function without modular arithmetic?

From what I understand, an elliptic contains a set points satisfying the equation $y^2=x^3 + ax + b$ together with the point at infity. It seems clear how multiplication with a scalar and a point ...
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128 views

Scalar multiplication of elliptic curve point by a fraction

I'm implementing an algorithm that works on a generic finite cyclic group written in the classic multiplicative notation: (G,*) = < g > , n = |g| At a ...