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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Finding the largest gap between the x coordinates of all points on an elliptic curve

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 ...
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463 views

Feasible attacks on ECRSA cryptosystem

In ECRSA cryptosystem, I want to know the feasible attacks. For illustrations we have two prime $p$ and $q$ such that $p \equiv 2 \pmod 3$, $q \equiv 2 \pmod 3$ and generate key pair as follows: $$n = ...
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213 views

Boneh/Franklin Identity based encryption with Tate pairing

Boneh/Franklin developed an identity based encryption scheme based on the Weil Pairing. This algorithm has also been standardised in IEEE P1363.3 . I know that this algorithm can also be implemented ...
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Can we break ECDLP with this machine?

Let $P$ and $Q$ are two points of NIST elliptic curve $E$ (defined over $F_{2^m}$ with prime $m$) and $k$ is a private key such that $k.P=Q$. Also we have a machine that is able to leak some ...
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281 views

How to compute projective coordinate Z in elliptic curve cryptography?

I was working on affine coordinates and struggling with the computation time taken for operations and then I was advised to use projective coordinates so that mul-inverse operation can be avoided Can ...
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61 views

Construct points with the same discrete logarithm

Assume we have an elliptic curve $E$ with a Tate (or Ate,...) pairing $G_1 \times G_2 \mapsto G_T$ Now the task is to find $g_1, g_1' \in G_1$ and $g_2, g_2' \in G_2$ such that the discrete logarithm $...
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350 views

RSA_DH vs ECDH implementation

In ECDH protocol is possible, naturally, to use the same algorithm for calculate a secret key for both communication parties (Alice and Bob for example). It is possible to design also a same algorithm ...
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119 views

Generating interactive, secure multiple ECC key pairs deterministically

In Elliptic Curve Cryptography (ECC) assuming user A has a private public key pair of $S$, $P$ accordingly with generator point $G$ which as we know is: $$P=S*G$$ Assuming that user A wants to ...
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344 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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205 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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103 views

Reliability of a single-pass deniable authentication protocol?

I look for one-pass deniable authentication protocol with a short message payload for my project and find a solution: ...
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359 views

Defending hybrid encryption schemes against padding oracle attacks

I intend to use a generic integrated/hybrid encryption scheme for transmitting information between a client and a server. Key encapsulation: a 128-bit symmetric key is generated and asymmetrically ...
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367 views

Known vulnerabilities in (EC-)KCDSA

Does anybody know if there's known vulnerabilities in KCDSA/EC-KCDSA? I have been researching for the past few hours and I haven't found anything. Wikipedia has very limited amount of information and ...
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549 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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46 views

GPG implementation of ECC “Encryption” (ECDH) vs RSA

My understanding of GPG with traditional RSA keys, is that RSA is by definition can be used to both sign and encrypt. This is because RSA can be directly applied to plaintext in the following form: <...
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35 views

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 ...
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93 views

Can the RSA accumulator scheme be converted to Elliptic Curve math?

Is it possible to translate the RSA accumulator scheme directly to EC without requiring bilinear pairings? In RSA we have: $A_{n+1} = A_n^c$ st. $\{c \: \textrm{prime} \: | \: c \in [\...
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83 views

Security of an Elliptic Curve Public Key with a “Small” x-coordinate

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 ...
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64 views

ECC with 512bit compatible curves

I understand that given solutions for solving a discrete logarithm problem are on the order of 𝑂(2𝑛/2), ergo, 256bit private keys based on 25519 or secp256k1 have an effective bit strength of ...
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31 views

short signature for EC

i'm building a low-power wireless sensor network in which each slave node has a public/private ECC key pair -- generated by the node itself during manufacturing.... the slave node is also provisioned ...
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1answer
27 views

Cost model for different curve models

Is there a cost model for each curve model and their conversions? For example: Take the curve models: Projective, Completed, Extended, Affine. Is there a table which shows how many multiplications, ...
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104 views

How do pairings behave on G2/twist points off the prime order subgroup?

$\newcommand{\F}{\mathbb{F}}$ Consider the ate pairing defined on a curve $G_1 = E(\F_q)$ and $G_2 = E'(\F_{q^r})$ where $E'$ is a twist of $E$ with the twisting isomorphism defined over $\F_{q^r}$. ...
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65 views

ECDSA - why is the first part of the signature used in the second?

Using the terminology of https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm Why is the second part of the ECDSA signature defined as: $s = k^{-1}(z+rd_A)\text{ mod n}$ ...
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41 views

Why doesn't the JOSE suite/JWA include ECIES?

The JOSE suite specifics use of RSA-OAEP (for when one party has an RSA key) and ECDH (for when two parties have EC keys) in JWA. Why doesn't it include ECIES? It seems like a way to derive a key ...
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51 views

Reason for including the public key of the key agreement in the KDF

I found the following text when looking up KDFs: In comparison, the so-called DHAES mode in IEEE 1363a mandates to use the binary representation of the sender’s public key as an input parameter. ...
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66 views

Patterns in elliptic curve division polynomials

While looking at division polynomials of elliptic curves in relation to this and this questions, I noticed some patterns. I am wondering if anyone knows of general formulas the describe these patterns....
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50 views

Is it necessary to transmit two or three points of an elliptic curve?

Are there cryptographic protocols, where a party should transmit by communication channel simultaneously two or three $\mathbb{F}_q$-points of an elliptic curve over a finite field $\mathbb{F}_q$? ...
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1answer
86 views

Authentication protocol for communication with Arduino Uno

I am using an ECDH key exchange to establish a shared secret between an Arduino Uno and an Android device. For this purpose I am using this library and more specifically Curve25519. This is the ...
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100 views

Security of BLS under additional information on the secret key

Question A Is the BLS signature scheme still secure if an adversary in addition to the public key $ pk = g_2 \, sk \in \mathbb{G}_2 $ also obtains additional information on the private key $ sk $, ...
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52 views

HW wallet and multisignaters in ECC

I need to design a system where there is a secure device (a.k.a. HW wallet), with the following functionality: Deterministic key generation for key parameters (speaking simply: key ID). Never expose/...
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53 views

What does the absence of Abelian group actions on supersingular isogenies implicate?

There are no Abelian group actions on supersingular isogenies. Why does this make them secure? - motivated by De Feo's Paper on mathematics of IBS
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43 views

Preserving location privacy

What are cryptographic techniques that could be used so that if I wanna to enable a server to send message to certain nodes in a network with preserving the privacy location for them ??
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54 views

What should I change in my implementation to pass from a curve $y^2=x^3+1$ to $y^2=x^3+x$

I tried to change my implementation of pairing in curves $y^2 = x^3 + 1$ to use curves of the type $y^2 = x^3 + x$ but it didn't work. I thought the only thing I had to change in my code was the ...
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292 views

ECC Curve25519: How to generate this kind of private key? / Strange key exchange mechanism

I'm currently reverse engineering a program that uses Curve25519 key exchange in network communication. I have only a basic understanding of ECC, so maybe this thing just seems strange to me. The ...
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123 views

Implementing key mapping across different elliptic curves

From this answer I understood how to prove key equivalence across two elliptic curves. Now, I'm trying to figure out some more practical aspects of implementing this. Before jumping into questions, ...
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113 views

Elliptical Curve Cryptography benchmark test

I have built web app that implement Elliptical Curve Cryptography. During testing using data set that i chose on my own, it already run well.. But, for now, i want to test it in some benchmark test, ...
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what does the m parameter in XML Signatures for gnBasis characteristic-two curves represent?

https://www.w3.org/TR/xmldsig-core/#sec-ECParameters defines the same three characters two-field basis's that http://www.secg.org/sec1-v2.pdf#page=107 defines: GN (Gaussian Normal, I guess) TP (...
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121 views

trying to understand the elliptic curve format for XML Signatures

https://www.w3.org/TR/xmldsig-core/#sec-ECKeyValue gives the following example of an ECDSA key: ...
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Can someone give me an idea of active areas relating to pairing based cryptography, and what they involve?

Apologies if this isn't the place to ask this question. I'm an undergraduate math student and I've been reading about elliptic curves. I've learned about the Weil and Tate pairings as well as Miller's ...
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What's wrong with this construction? ECDHE+AES

Let's say I want to use ECs to do asymmetric encryption. Suppose both sides have generated EC keypairs and have exchanged their public keys. I run ECDHE, and get a derived secret. I would then either ...
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206 views

An efficient SPEKE protocol with curve25519

SPEKE is a very simple and elegant PAKE protocol. I think, one of the reasons why PACE has been invented and is now the ICAO protocol, is because SPEKE was patented. Fortunately, the SPEKE patent ...
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How can I split a packed Ed25519 public signing key into its X and Y coordinates?

I'm using the Curve25519 code (from http://www.dlbeer.co.nz/oss/c25519.html), and trying to convert from a public signing key (Edwards form) to a public key-exchange key (Montgomery form). There's ...
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how researchers extract encryption times when randomness is involved?

I have seen some research papers where they compare encryption/signature times on elliptic curves.But when i have implemented the algorithms and calculated encryption times but each time i get some ...
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947 views

Is ECDH(E) Key Exchange FIPS 140-2 compliant?

We have read dozens of documents now - some that contradict each other - and cannot find a solid source of truth. Does FIPS 140-2 compliance allow for the use of elliptic curve cryptography as a key ...
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96 views

Lower bound for the size of prime factors?

We all know classic RSA and that we should pick moduli of at least 2048-bit length to get decent (112 bit) security. Now there's also multi-prime RSA, which can yield significant speed-ups using the ...
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152 views

Isomorphic curve vs isomorphic curve group

It is said that if two elliptic curves $E_1, E_2$ defined over a finite field $K$ are isomorphic, then $E_1(K), E_2(K)$ groups are also isomorphic. But the converse is not true. Now, By definition, ...
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262 views

How Elliptic Curve equation is chosen?

We all know the basic equation of Elliptic Curve is $y^2 \equiv x^3 + ax + b \pmod p$ How the value of the constants $a$ and $b$ are chosen? Suppose $\mathbb P\ni p \approx 2^{256}$ then what ...
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Is discrete log as hard when given same challenge in both pairing groups, in the case of BN curve and its sextic twist

This is a special case of the question Generalization of the DL-assumption in bilinear group pair, that wasn't answered. Suppose $G_1$ is a BN curve over $F_q$. That is the set of elements $(x,y)\in ...
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88 views

Caculated time One Point Multiplication with double and method

I am using double and add method for point multiplication in affine coordinates. How we compute 1PM in double and method? Di Wang said one point multiplication consists of repeated addition and ...
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241 views

Example of Projective Coordinates

Given the affine form of coordinates $(x,y)$ such as $(5,3)$, if I want to convert $(5,3)$ to projective coordinates $(x,y,z)$, should the form of point be $(5,3,1)$? It is triplet not a point, right? ...