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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Do Weak Elliptic Curves Exist?

I'm running into very contradictory opinions when try to understand if weak elliptic curves exist. I'm not interested in the case when a curve's weakness is attributed to properties of an EC's prime, ...
4
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1answer
99 views

Are EC public/private keys significantly weakened by having a known byte?

I would like to use different EC keys for different purposes in an app, and I would like to easily see the purpose for which particular key (pair) was generated. With ~65K attempts, I can generate a ...
2
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1answer
74 views

Multiplication in elliptical curve cryptography

I'm practising for a cryptography test and came across this question: Let us assume that for a specific elliptic curve the formulas for the computation of $(x_3, y_3)=R+S$ are given by: $X_3=...
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1answer
92 views

All the affine points on the curve

I have calculated the affine points on the curve $x^2 + y^2 = 1 − 3x^2y^2$ over the field ${\mathbb{Z}}_{11}.$ Using $y^2 = \frac{1-x^2}{1+3x^2}$ I got the following points: $(0,1),(0,10),(1,0),(2,2)...
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312 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 ...
2
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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 ...
3
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1answer
73 views

How to compromise the privacy of NaCl encrypted messages, when nonce is reused?

I want to know to how break NaCl encryption when a nonce is used more than once for a given key private key. According to PyNaCl docs: It is VITALLY important that the nonce is a nonce, i.e. it is ...
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13 views

Transforming EC public key X and Y-sign into X and Y [duplicate]

I'm using bouncycastle which uses X and Y coordinates for public keys. An EC public key is more compactly represented by the X coordinate and the sign of the Y coordinate. How do I use bouncycastle ...
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319 views

How does Diffie–Hellman differ from elliptic curve Diffie–Hellman?

I didn't understand how ECDH actually works. Disclaimer: I know very little about elliptic curves. Here is how DH works: Alice and Bob agree on a prime number $P$ and a generator $G$. (They use one ...
3
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1answer
108 views

Can a billion elliptic curve keys be generated on a laptop in less than an hour?

I want my application to generate an EC key pair. The first four bytes of the sha256 hash of the public key should contain a known IP address. As hashes are one-way functions, I need to brute force ...
3
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1answer
304 views

ECDH or RSA more secure for symmetric key wrapping?

Suppose a message is encrypted with a symmetric block cipher with a random key. RSA is often used to wrap the symmetric key using the recipient's public key. In this case, the size of the message is ...
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56 views

Using a product of a series of curve25519 scalars as a private key

There are a few systems like the GNU Name System and the Sphinx mixnet packet format that employ a series of curve25519 scalars all multiplied together as a private key. Are there any caveats to ...
6
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1answer
113 views

Encoding scalar values to points on Ed25519

I'm interested exploring key derivation and threshold signature protocol that require point arithmetic (addition) on the private scalar values and $S$ values of the signatures in ed25519. ...
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1answer
100 views

About end to end algorithm SMS from Jo Mehmet Sollihagen Øztarman [closed]

This is a signcryption scheme from End-to-End Data Protection of SMS Messages by Jo Mehmet Sollihagen Øztarman (pdf, section 4.3.1): Public parameters C: an elliptic curve over GF(ph) with p ≥ 2^...
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50 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 $...
6
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1answer
116 views

How to exctract ECDH parameters from an OpenSSL-generated $G$?

I'm using ECDH for generating ECDH public parameters (p,a,b,G,n), I try to get this values using openssl ecparam -in cert.pem -text -noout For ...
3
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1answer
111 views

Do ECDH and ECDSA combined solution provide an authenticated protocol exchange solution in MCU environment?

I'm working on a project in MCU (micro controller) environment with GPRS connection to exchange data securely with the cloud. The encryption algorithm has been selected to use AES. I did some search ...
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2answers
142 views

Determine if a public key point y is negative or positive, odd or even?

Take an elliptic curve cryptography public key (x, y) and its additive inverse (x, -y). How do you identify which is the positive point and which is the negative point? Examples: Private key 1 -> (x,...
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3answers
156 views

Safe curves in Weierstrass form?

I would like to implement a protocol using elliptic curves. I'm thinking of using MIRACL so using curves in their Weierstrass form is preferable as it they are supported by this framework. I don't ...
5
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0answers
67 views

Elliptic curve point addition with $Z_1 = Z_2 = 1$

Elliptic curve point addition and point doubling operations using Projective and Jacobian coordinates require fewer field multiplication operations when considering $Z$ coordinates of input points ...
4
votes
1answer
206 views

Why a key pair is used in OpenSSL ECDSA_sign or ECDSA_do_sign instead of only private key ?

If the digital signature in ECC (ECDSA) is made using a private key, why is the OpenSSL library using ECKEY pair as an input parameter instead of the private key only ?
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383 views

Forward-secure static-ephemeral ECDH key agreement protocol

The question is whether the following simple key agreement protocol design has good security properties, and how it can or should be improved. Assumptions Alice is a persistent entity with a static ...
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34 views

Where can I find a versatile PKI and library leveraging curve25519?

I note that Things that use Curve25519 provides a long list of protocols, software, and libraries that leverage curve25519, but I'm wondering if there is any such software that more fully implements a ...
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51 views

What is the hardness in Decisional Linear Assumption (DLIN)?

I had understood what does the DLIN assumption means and here is a related question. But I fail to understand the 'real hardness' in this problem. I would be grateful if someone can help me to ...
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49 views

Could someone explain the given algorithm?

The following snapshot of the algorithm is taken from the book Guide to Elliptic Curve Cryptography published by Springer 2004. I don't understand that why at the statement 9.1 if both $T_1$ and $...
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1answer
149 views

ECDH and ECDSA in PGP with known public key

How are used ECDH and ECDSA in combination with public key ? Usually these methods are used for establish a secret themselves, ...
4
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1answer
116 views

OpenPGP Public-Key algorithm

What means when pgpdump shows this kind of algorithm in the Public-Key Encrypted Session Key Packet? ...
11
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1answer
651 views

How to calculate elliptic curve parameters?

I'm having a rough time understanding the math behind elliptic curves. I want to implement ECDH where user can define a, b, and p parameters of elliptic curve. How can I calculate generator base ...
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2answers
359 views

How to derive a symmetric key from ECDH shared secret?

I am trying to implement the internal primitives of ECDH. Currently I'm able to multiply the receiver's public EC point with the sender's private key to arrive at the shared EC point. Next step is to ...
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2answers
460 views

How to determine the order of an elliptic curve group from its parameters?

Let $\quad E:\; y^2 = x^3 + ax + b \quad$ be an elliptic curve defined over a finite field $\mathbb F_q$ where $q = p^n$, $a,b \in \mathbb F_q$ and $p \neq 2, 3$. By Hasse's theorem we know that the ...
2
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1answer
206 views

OpenSSL ECDH key exchange mechanism

I am using FIPS based OpenSSL module for encryption of sensitive data for my desktop socket server and client applications. I am using ECDH for key agreement.The keys public and private pair is ...
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1answer
310 views

Why Curve25519 for encryption but Ed25519 for signatures?

NaCl and libsodium libraries use Curve25519 for authenticated encryption (actually for sharing a key which is used for encryption) and Ed25519 for signatures. What is the purpose of using different ...
2
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1answer
307 views

What are the differences between curve NIST P-521 and Edwards E-521 for signature?

I implemented and used the P-521 curve for ECDSA. Signatures are 132 bytes long. It seems that Edwards E-521 is safer but I did not investigated very deeper. What is its signature length ? How is it ...
12
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1answer
248 views

Edwards / Montgomery ECC with Weierstrass Implementation?

So let's assume I want to perform Ed448 or Ed25519 digital signatures or want to perform a DH key-exchange. Assume further that those curves (Curve448 or Curve25519) are required. But the problem is, ...
5
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2answers
454 views

Is Curve25519 vulnerable to private key exposure in the case of a bad RNG?

I'm really excited by what I've learned of advancements in elliptic-curve cryptography. Curve25519 seems to be a great choice at this point in time, but if I recall correctly, some elliptic curve ...
11
votes
1answer
560 views

Does Curve25519 only provide 112 bit security?

In a recent mail on the IETF CFRG mailing list it was claimed that: The (currently missing) security considerations (or somewhere) should describe why Curve25519 is ok when used in contexts where ...
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33 views

Safe generation of $k$ points on a curve such that the mutual discrete logs are hard?

I have a multiplicative group $G$ of prime order $p$ implemented using a twisted Edwards curve (similar to Ed25519). I want to compute a set of $k$ distinct points $P_1,...,P_k$ that generate $G$, ...
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3answers
189 views

Supersingular Isogeny Key Exchange broken?

Found this report detailing a quantum algorithm for computing isogenies between supersingular elliptic curves. http://cacr.uwaterloo.ca/techreports/2014/cacr2014-24.pdf with the quote "...
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1answer
169 views

Modular reduction for NIST prime P256— understanding the data

I am working on a project where I need to implement elliptic curve cryptography, I am struggling from a long time in order to understand the working and the process. Modular finite field arithmetic, ...
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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 ...
4
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3answers
142 views

Calculating $\mathbb F_{p^2}$-rational points of an elliptic curve defined over $\mathbb F_p$

How can I calculate points on an elliptic curve defined over $\mathbb F_p$, for example $y^2 \equiv x^3 + 1 \pmod p$, with coordinates in $\mathbb F_{p^2}$? (points might have complex number format in ...
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0answers
45 views

Testing PRNG quality from ECC public keys?

Having a large set of ECC public keys $P_i = n_iB$ on a fixed curve $E$ over a prime field, is there a way to determine if coefficients $n_i$ were generated using a bad PRNG? In other words, can a ...
5
votes
1answer
115 views

Why not to use curve over field of $p^m$ with $p > 2$ for ECDSA?

I'm reading the ECDSA paper and they say you can only use ECDSA with odd-power fields $p$ or with binary fields $2^m$. Why not other power prime fields?
3
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1answer
252 views

public key cryptography and authentication for cross-platform network application

I'm developing network based application for control and telemetry on Linux based embedded system. I'm using ZMQ network library and Google Protocol Buffers serialization library for communication ...
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1answer
246 views

Is SHA-1 safe for signing ECDHE parameters?

Is using the SHA-1 algorithm insecure for hashing the ephemeral ECDH public key in the signed_params structure? There are some worrying articles about using SHA-1: ...
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1answer
175 views

Base point in Ed25519?

The paper "High-speed high-security signatures" by Bernstein et al. introduces the Edwards curve Ed25519. Concerning the base point $B$, it says that $B$ is the unique point $(x, 4/5)\in E$ for ...
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145 views

Attacks on elliptic-curve based cryptosystems through solving the Decisional Diffie-Hellman Problem with the Weil Pairing

Are there any examples of practical attacks on cryptosystems set over elliptic curves which utilize the easiness of DDH for certain choices of curves $E(\textbf{F}_q)$, and as such their lack of ...