# 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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### Size of group for Elliptic curves vs RSA for equal security

For my research, I would like to compare the efficiency of a scheme when instantiated with Elliptic curves and RSA. So, I would like to know a "latest" comparison (as of 2018) on what group sizes of ...
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### ECDSA signing and verification between Python and JS [closed]

I'm trying to have Python (2.7) and JS solutions for ECDSA signing (with secp256k1 curve) where ideally signatures generated by one side can be verified by the other. For the python side, I'm using ...
451 views

### Compact encoding of an elliptic curve point

I'm working on a project with elliptic curve cryptography (ECC), I'm using the secp256k1 library (the one that's used in bitcoin). My goal is to create the most compact platform-independent ...
325 views

### Encrypt or decrypt using the private key, and decrypt using the public key?

I am currently trying to figure out if my following scheme is implementable in ECC or whether I can use existing implementation. At first I was going with libsodium but it seems that it really doesn't ...
375 views

### How is the precomputed table for 25519 Elliptic curve generated?

I am wondering how the precomputed table for scalar multiplication for elliptic curve (in my case 25519) is generated/precomputed? I am talking about this [https://github.com/WhisperSystems/...
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### Number of generators of an elliptic curve

Consider the elliptic curve E:$y^2 = x^3 + 3x + 11\,\, mod\,\, 19$. Two questions: Let the cardinality of the set of points on the elliptic curve( including $O$ ) be $|E| = 25$. How many points are ...
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### Do the elliptic curves over prime fields must always contain prime number of elements (prime order)?

I have gone through one example where i saw a curve defined over some prime number containing non-prime order.
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### 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 is R(-25,30i) where is sqroot -1 Chosen two ...
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### Why is RSA still being used? [duplicate]

In hybrid encrytion, I still see that some site's use RSA in their https connection, so now I wonder, why do they not use ECC instead of RSA, ECC requires less computational power and encrypt's and ...
267 views

### Proper forward secrecy [closed]

Currently I have a protocol using a simple RSA to AES handshake. I have been reading more and more and would like to implement proper forward secrecy, but at the same time I'd like to improve the ...
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### 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 ...
126 views

### Strength of a cryptography algorithm [duplicate]

I'm just read this article from Atmel corporation in comparing RSA with ECC cryptography algorithms. First of all please read these two paragraphs quoted from the article: P1: Strength of an ...
231 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 ...
481 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 ...
200 views

### becoming a cryptographer after math studies [duplicate]

After studying philosophy and being a philosophy teacher, I took back studies 4 years ago and I did a bachelor in maths. I'm in maths grad school now (I'm 32), and I would like to work in cryptography....
214 views

### An elliptical curve over GF(2^3) is defined as y^2+xy=x^3+ax^2+b with the given value of a= g^3 and b=1.R = P + Q, where P = (0, 1) and Q = (g^2, 1)

An elliptical curve over $GF(2^3)$ is defined as $y^2+xy=x^3+ax^2+b$ with the given value of $a= g^3$ and $b=1$. $R = P + Q$, where $P = (0, 1)$ and $Q = (g^2, 1)$ Can someone solve this question ...
322 views

### ECC public key encryption without symmetric cipher

Imagine the following scenario. A process is running in background and permanently encrypting some data. An adversary has full control of the process, e.g. it can dump the process memory any time and ...
160 views

### Is this the right way to implement ElGamal scheme over Elliptic Curves over prime field? [duplicate]

I'm fairly new to Cryptography, especially elliptic curves in general. I learned to do Point Multiplication, Scalar Multiplication and also programmatically implemented them. But I was trying to do ...
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### What are the inverse operations in elliptic curve cryptography?

Public-private key cryptography is based on inverse operations that use separate input. In elliptic curve cryptography, what are those inverse operations?
510 views

### Elliptic Curve Cryptography calculation of $y^2 \equiv x^3 + x + 1 \pmod{23}$

Learning the basics of elliptic curve cryptography. The question is a mathematical one. While finding the points in the elliptic group $E_{23}(1,1)$,this is how one proceeds : How is $y^2= 7$ ...
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### Equivalence of cryptographic problems

Are integer factorization, discrete log and ECDH problems equivalent? I know that factorization and discrete log are equivalent but are one of those two problem equivalent with ECDH? Cand someone ...
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### ECDH over secp256k1 implementation

I'm using ECDH over secp256k1 in multiple languages, and I saw something a little weird in the rust library I use. After the point multiplication it changes y to <...
178 views

### Getting coefficients of Curve25519

I want to extract coefficient $A$ and $B$ from Curve25519 represented in Montgomery form as $B v^2 = u^3 + A u^2 + u$. What are the $A$ and $B$ coefficients of Curve25519 in Montgomery form?
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### What are the steps for finding points on finite field elliptic curves?

New to the crypto world and I can't figure this basic thing out, embarrassingly. Bitcoin uses secp256k1's elliptic curve y^2 = x^3 + 7 mod(p) Let's pretend p is 9. Using this little website: https:/...
309 views

### Abelian groups in Elliptic curves [closed]

Do every elliptic curve defined over a prime field forms an abelian group?
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### Elliptic curve points addition is not associative [closed]

I've found an article that says how to add points in projective coordinates.But in my implementation these points don't form a group. Fields: ...
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### How to convert (Rx1 and Ry1) to (Rx2 and Ry2)

I'm working with the secp256k1 elliptic curve and have point doubling and point addition formulas for this curve. If a point is given $Q_x$ and $Q_y$ ...
108 views

### How to find "k" in system of equations?

This is a $y^2=x^3+7$ elliptic curve points - $Q,G_1,G_2,G_3. k_1,k_2,k$? - secret exponents: $k_1*G_1( x_1,y_1) = Q(X,Y)$ $k_2*G_2( x_2,y_2) = Q(X,Y)$ $k*G_3( x_3,y_3) = Q(X,Y)$ How to find a $k$?...
119 views

### Points on elliptic curve [closed]

I am making a program using the library cryptopp using curve secp521, in which at the end of that program I get n*Point Because I am writing that program I know that what is the value of 'n'. So, I ...
901 views

### ECDSA public key generated with constant prefix?

I have some javascript, which generates new ECDSA public-private keypair. However all the resulting public keys which I generate seem to have fixed prefix. This seems strange to me, since the ...
3k views

### How to perform the Elliptic Curve calculation in the following example?

can someone show me the working how to get (10,6) what i am getting is (10,5) for 3P
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### elliptic curve point doubling in Jacobian coordinates

I am writing an application that uses Elliptic curve Diffie–Hellman for authentication. I found two formulas for point doubling in Jacobian coordinates. 1st) \begin{equation} X_1 = (3x^2 + aZ^4)^2 -...
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### Why don't secp256k1 use a prime order subgroup?

Using a prime order subgroup prevents mounting a Pohlig–Hellman algorithm attack. Meanwhile, secp256k1 doesn't use a ...
159 views

### Can the public key be derived from the private key? [closed]

The calculation/formula i use in deriving a public key from the private key without importing any module in python3 script involves the following steps: Define the parameters of the secp256k1 ...
81 views

### 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 ...
582 views

### How to use public/private keys in elliptic curve cryptography to encrypt/decrypt information

I'm reading a bit about elliptical curve cryptography. The basic idea is to define a dot-operator on the points of an elliptic curve. Given a starting point $P$, and applying this dot-operation $n$ ... 92 views

### Which elliptic curve cryptography algorithm can be used for this scenario and how to do? Please explain step by step

I want to implement the below scenario using elliptic curve cryptography: Steve has list of IDs and he wants to encrypt with private key - IDs Steve send the list of encrypted IDS to John - Steve(...
170 views

### Understand new fast computation algorithm elliptic point

I'm reading about new fast computation algorithm elliptic point in this paper Analyzing the Point Multiplication Operation of Elliptic Curve Cryptosystem over Prime Field for Parallel Processing. But ...
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 $... -1 votes 1 answer 168 views ### How Proof all Subgroup point of prime order elliptic curve have prime order [#G=#E]? [closed] anyone knows any reference that proof it ? Please Help . -1 votes 1 answer 263 views ### counting points not on elliptic curve [closed] Given an curve with equation$y^2=x^3+ax+b$, I want to find the number of pairs$(a,b)\in \mathbb{F}_p \times \mathbb{F_p}$NOT on the curve. How do I do it? I have an intuition that it is$p$, but ... -1 votes 1 answer 112 views ### Is size Q equal to size SHA(Q)? [closed] Assume d is a 128 bit random integer and P is base point of an elliptic curve and Q = dP is a point on the elliptic curve and SHA is a hash function with 128 bit output, my question is: Is size Q ... -1 votes 1 answer 101 views ### The benefit of the shared key generated from ECDH key exchange protocol [duplicate] What the benefit of the shared key that generated from ECDH key exchange protocol? Can I use it to encrypt the message or encrypt the public key of asymmetric encryption? -1 votes 1 answer 218 views ### Breaking Elgamal private key I have been given a task to explain if - given a public key and a portion of a private key (over 300bits) with a remain unknown of 80 bits - the private key can be broken with an algorithm faster than ... -1 votes 1 answer 384 views ### Is that right what I understand about MOV attack? There is an elliptic curve.$y^2 = x^3 +ax+b \pmod p$($p$is prime number) To solve DLP, need to find$r$from given points$G$,$rG$. ($G$'s order is$q$and$q\$ is prime number) The MOV attack ... 