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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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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56 views

Efficiently formatting data so that each byte is critical

I'm experimenting with implementing a Fair Exchange Protocol that assumes there is a trusted third party to provide a hash-checksum of the two files for each missing byte, (so that the exchange can ...
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236 views

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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183 views

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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510 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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229 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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164 views

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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295 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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381 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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122 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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358 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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206 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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105 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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364 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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368 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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569 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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43 views

Elliptic Curve (Point Counting)

I am studying elliptic curves in particular point counting. If I have coordinates P and 2P, is there a way to calculate the total points in between P and 2P using either curve parameters or algorithm? ...
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55 views

is forward secrecy irrelevant for non-streaming applications?

I asked a question yesterday about the Keybase key model and got no answers, unfortunately. Let me rephrase the question to make it clearer: in the case, if 2 users just want to send each other low-...
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40 views

Trying to understand Keybase's key model and replacing PGP with device keys

I am exploring Keybase and I thought it was merely a wrapper for gpg and connecting its public key with social accounts (e.g. github, twitter, etc...). But after reading the very short and unclear ...
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47 views

Verifying the ownership of curve25519 public keys

Let's say we have a group of users, authenticated by a server that providers the service, communicating on a secure channel (e.g. over HTTPS/TLS) and each user has a corresponding curve25519 key pair. ...
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1answer
66 views

Complexity of computing zk-SNARK Proofs

Disclaimer: I have no background in cryptography, and everything I'm asking about is what I've learnt from last couple of days of frantic reading on this topic. Any help is much appreciated. Q: What ...
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38 views

Need help understanding SPAKE2 setup values

I am trying to write a simulation of the SPAKE2 protocol in python (just so I can get a better understanding of the protocol altogether). I am reading through the ietf draft here: Datatracker. There'...
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49 views

Using ECC CDH test vectors with ECDH when h >1

I am writing formal tests for a system with a number of crypto requirements including support for ECDSA, ECDH and HMAC. The system is required to support the following EC's: NIST curves P-224, 384, ...
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69 views

How to find kernel of isogeny from the dual isogeny

Let $E$ be a supersingular elliptic curve over $\mathbb{F}_{p^2}$, where $p = \ell_A^{e_A} \ell_B^{e_B} f \pm 1$ for some primes $\ell_A, \ell_B$. Let $R \in E[\ell_A^{e_A}]$ be a point of order $\...
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56 views

Elliptic curve discrete logarithm problem

I'd like to know what is the maximum bits of the finite field that we can solve the ECDLP in a "regular" computer in trivial time. Is there any recent data about this?
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43 views

EC threshold private key's multiplicative inverse and derived-key sharing

I have two devices, and each has a private key xPriv-i. Each device computes the corresponding EC public key xPub-i, shares it, and the linear combination of the keys is the "real" public key xPub. ...
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54 views

How to know if a point on a discrete elliptic curve be represented uniquely using its y-coordinate?

Let's say we have a point on an elliptic curve $p=(x, y)$ which is not the point-at-infinity. Can there be some other point $\hat{p} = (\hat{x}, y)$ that is also on the curve and that has the same y-...
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1answer
45 views

Ensure Data Integrity In An ECDH Key Excange

Been playing around with the inner workings of onion routing and I have a problem. If I wanted to send the 2nd node of a relay network an ephemeral ECC public key, it has to go through node 1, so that ...
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125 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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53 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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1answer
91 views

Difficulty of Reversing Elliptic Curve

In ECC, it is apparently easy to verify the final point given the starting point and the number of hops. But it is difficult to compute the number of hops given just the starting point and the final ...
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129 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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41 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
42 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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112 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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50 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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61 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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76 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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52 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
117 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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109 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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56 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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54 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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330 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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132 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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126 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, ...