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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Regarding the isogeny path problem

Given two elliptic curves, it is hard to calculate an isogeny of large degree between them. Does this only apply to supersingular isogenies or to ordinary ones as well? Additionally, is the mapping ...
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60 views

Isogeny of elliptic curve

If we have two elliptic curves $E$ and $E'$ and the points of both elliptic curves are same. Then all the points of $E$ map to all the points of another elliptic curve $E'$. For example $E$ has ...
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How does the order of a group, it's torsion subgroup and the co-factor link?

Given an elliptic curve that defines some group of non-prime order, with co-factor h. Would it then have a h-torsion subgroup? What are the implications for ECC ...
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Differential representation of binary curve as a public key

Given elliptic curve $E$ over binary field $k$, a public key is the pair $(x,y)$ in $E$ and $x$ and $y$ in $k$. The differential representation of $(x,y)$ is $w = x + y$. What security implications ...
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Using division polynomials to prove that EC discrete log is even

This question is related to the other question I recently asked. I'm trying to figure out if it is possible to use division polynomials to prove that knowing $A = a \cdot G$ we can prove that $a$ is ...
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114 views

Sharing secret content with multiple recipients

I have a sender and N recipients, and am thinking of using the following scheme to send secret content to those recipients. This is similar context to a group chat or email. I am no expert in crypto ...
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50 views

Algorithm to compute DLOG for elliptic curve $E(F_p)$ with order p

I was reading about elliptic curves in this pdf. Page 55 of the pdf states that if number of points on elliptic curve #$E(F_p) = p$, then there exists a p-adic logarithmic map that homomorphically ...
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NIST ECC Curves without pairings

NIST FIPS.186-4 has standardized 5 ECC curves on field $\mathbb{F}_p$ (P-192, P-224, P-256, P-384, P-521) and 10 elliptic curves on binary fields. I need to use ECC curves without pairings for my ...
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292 views

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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190 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 ...
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154 views

MTI/A0: modular arithmetic or elliptic curves?

In the Handbook of Applied Cryptography (Menezes, A. J.; van Oorschot, P. C. & Vanstone, S. A.) Protocol MTI/A0 key agreement (algorithm 12.53) described as $\mod p$-protocol. The survey Overview ...
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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[5] is R(-25,30i) where is sqroot -1 Chosen two ...
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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 ...
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Is there any patent free EC point compression available? If not, should I GPL/LGPL my code?

I'm currently using the BouncyCastle implementation in uncompressed mode, and considering releasing an optimized version of EC code that uses point compression. It appears the only thing limiting me ...
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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 ...
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367 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 ...
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86 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 ...
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Generate keys with Edwards25519 - Equivalence to Curve25519

I am trying to generate Ed25519 keys based on a 32-byte seed. According to RFC 8032, in order to create the public key, you have to take the private key, hash it with SHA-512, prune it, and then use ...
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ElGamal ECC. and Menezes-vanstone ECC

Can anyone tell me how to measure the security force of ElGamal ECC. and Menezes-vanstone ECC.?
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397 views

Elliptic curve discrete logarithm problem and mini example

Consider the group E23(9,17), this the group defined by the equation y2 mod 23 = x3 + 9x + 17 mod 23. What is the discrete logarithm k of Q = (4,5) to the base P = (16,5)? the solution is: 2P = (20,...
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61 views

Simple hexadecimal to 2s complement question

From 2.5.1 in this paper, how is $p$ = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFFFF 00000000 00000000 FFFFFFFF = $2^{384} − 2^{128} − 2^{96} + 2^{32} − 1$ ...
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Table of Curve Paramters

I'm studying Elliptic Cruve Cryptograhpy. When I do a search on Google of ECC, I find some pdf where I see these curve's paramters: $q, h, r, exp1, exp2$. What are these parameters ? Are there tables (...
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151 views

Can an elliptic curve have the form y^2 ≡ x^2 + 2x + 2 mod 17?

I'm new to cryptography and the associated level of maths. I'm practising past papers for an exam and found the question: Show that the condition 4a^3 + 27b^2 ≠ 0 mod p is fulfilled for the ...
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if ABE scheme based on RSA generate constant secret key will get better resutls than schemes based on Elliptic curve?

We know in term of security level, ECC key is stronger than RSA , 160 bit ECC equivalent to 1024 bit key in RSA. If I can generate constant secret key based on RSA is that mean my scheme performance ...