# 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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### How can I understand the math behind crypto, especially that used in blockchains / distributed ledgers?

A bit of a background about me; Computer Engineer but during my studies I didn't dive too deep into the maths, especially the crypto maths that is used to make the blockchain and cryptography work. I'...
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### How to sign CAM/DENM messages using mbed LTS libray with ECDSA (EEE Std 1609.2)

I have read ETSI TS 103 097 and EEE Std 1609.2 which gives me the security data structure defined in ASN.1 notation. I want to use mbed LTS to calculate the ECDSA signature. I understand that I should ...
257 views

### What is a function on a Line or a Curve?

I am reading up on Pairings using Elliptic curves & all the texts talk about functions on a Curve. I am finding it difficult to even figure out what they mean by "function on a curve" or ...
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### AWS IoT - Unique Keys per each device - Data encryption

(Probably not the right board to ask. But here goes) I'm designing an IoT Solution with RPi as client and AWS as the server. On the client hardware, I have an Security Chip that can securely generate ...
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1 vote
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### computation time of pairing operations and their securities

Suppose G1 is an elliptic group and G2 be a multiplicative group and they are of same prime order p and e is a bilinear pairing, e: G1 X G1 -> G2. The operations e(p,q)r and e(pr,q) gives equal ...
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### Short Nonces in ECDSA signature generation

Recently I noticed that my device generates short-sized Nonces. Approximately $2 ^ {243} - 2^{244}$. Could it turn out that there will be a small leak of ...
1 vote
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### Convert secp256k1 private key to sr25519 private key

Is it possible to convert secp256k1 private key to valid sr25519 key?
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### Would being able of factoring integers efficiently have some consequences over Elliptic Curve Cryptography?

Let's assume you can factor integers in a very efficient manner. Would that endanger the security of e.g. elliptic curve cryptography, or is there no link between the two ? You can often read that ... 170 views

### Two Elliptic Curve Points having the Same X coordinate

Suppose in a elliptic curve (say the curve equation is: $y^2 = x^3 -17$) with prime order $q$, we have $(x,y_1) = nP$, where $P$ is a generator and $n<\lceil{q/2}\rceil$. Can we claim that there ...
1 vote
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### Birational transformation from Edwards curve with not square d to Edwards curve with square d

How can I transform a complete twisted Edwards curve $ax^2+y^2 = 1+dx^2y^2$ with not square $d$ and square $a$ into an isomorphic Edwards curve $X^2+Y^2 = 1+DX^2Y^2$ with a square $-D$ i.e. $D = -r^2$?...
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### Is it possible (and if so how) to make one proof for multiple private keys in ECDSA

Lets say I have a message that needs to be signed by two keys that were generated using ECDSA Is it possible to make a signature that accounts for both keys, meaning I can verify with both and see ...
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### ElGamal with elliptic curves and semantic security

To encrypt a group element $P$ with public key $K$ and randomness $r$ using ElGamal on elliptic curves with base point $G$ we do the following $(c_1, c_2) = (r\cdot G; P+r\cdot K)$. When we want to ...
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### Are there any public keys for which the private key can be easily derived (ECDSA)?

I know that generally it's infeasible to find the private for any given public key. But I also came across the question "Find ECDSA PrivKey to PubKey = 0", in which it was explained that the ...
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### Can I know from a Bitcoin public key if the private key is odd or even?

Can I know just from a Bitcoin public key if the private key is odd or even? [moderator note] That is, can we find parity of the private key from a secp256k1 public key?For the original dump of ...
1 vote
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### running Project Wycheproof against crypto implementations in languages other than Java

So I guess https://github.com/google/wycheproof "tests crypto libraries against known attacks". It appears to mainly be intended for Java crypto providers but can it easily be adapted to be ...
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### What is/was SEC#1 ECC public key leading octet 0x01 for?

In the SEC#1 elliptic curve cryptography standard, the encoding of the public key involve a leading octet: 00h: The public key is the point at infinity. 02h, 03h: The public key is the compressed ...
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### Why does ECDSA produce a pair of values in its' signature (r,s)?

I was wondering why ecdsa generates a signature in form of a pair (r and s) and why it can't be only one value.
1 vote
1k views

### What is the difference between "Elliptic Curve Function" and "Hash Functions" like SHA256?

I am reading about bitcoin and I am a little confused about "elliptic curve function" and "SHA256". Do they have the same properties? Can both be used to generate private and ...
1 vote
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### Multiuser encryption, singleuser decryption [duplicate]

I have an hybrid encryption (RSA, AES) for a file sharing project I am working on, where I use a single public key for encryption on the sender side and corresponding private key for decryption on the ...
1 vote
215 views

### ECDSA - generating a new private key each time we sign?

So, I kinda get the mathematics behind the ECDSA, but I can't seem to find precise information about private key generation. In other words, do we have to generate private key, each time we generate a ...
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### Size of group elements in a bilinear context

In a asymetric pairing context, which size (in bits) should have the elements of $\mathbb{G}_1,\mathbb{G}_2$ and $\mathbb{G}_T$ if we consider the most efficient elliptic curves?
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### What is the meaning of $F_{p^k}$ and the elliptic curve over it, $E(F_{p^k})$?

In pairing based cryptography, there will be the finite field $F_{p^k}$ where $p$ is prime number and $k$ is an integer. The elliptic curve is constructed on that finite field as $E(F_{p^k})$. For ...
1 vote
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### Isomorphic mapping of BLS12-381 G2 points to G1

I'm attempting to reproduce ring signatures as described in Section 5 of https://crypto.stanford.edu/~dabo/pubs/papers/aggreg.pdf but applied to the BLS12-381 system. One of the assumptions in their ...
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### kleptography SETUP attack in ecdsa

I'm trying to implement kleptography SETUP attack of ecdsa with python. Just a simply script to verify the algorithm. However i can't get the right output as the paper said. Where is the problem? Can ...
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### How long does it take to generate signature for Elliptic Curve keys using the P-256 curve?

If you have a plain text document, known public key to verify generated signature strings against. EDIT: You do NOT know the private key, this is all you have. Using a modern computing power with 4 ...
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1 vote
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### OpenSSL EC PRIVATE KEY content structure details

Background I am trying to understand how PEM contents are formatted for "EC Private Key" so e.g. following is private key ...
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### Is it possible to get the x point of the secp256k1 elliptic curve knowing only the y point

There is a list where, using the coordinates of the x points, it was determined whether there are points in the curve Here's a link It can be seen that the generator according to the formula ...
1 vote
181 views

### Q about points on an ECC curve

I'm trying to learn about ECC. I understand that the points of the finite field are determined by taking the continuous elliptic curve and finding its points that have integer coordinates. Since ECC ...
1 vote
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### The significance of the field of the factor in Lenstra’s ECM

I am going through Lenstra's Elliptic Curve Factorisation from Silverman's Mathematical Cryptography book. I have understood the algorithm itself, but unable to understand a specific point the book ...
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### How to find integer point of a ec curve in a given range?

I was looking inside the basics of ecc and found the examples from Internet either uses continuous domain curve or use a very small prime number p like 17 in a ...
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### Is it safe to reveal an arbitrary EC point multiplied by a secret key?

We have a primary-order EC group. Need to perform a (sort-of) DH protocol, whereas the key is permanent, not a nonce (single-usage ephemeral key). So we receive an arbitrary group element (EC point) ...
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