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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Prevent MITM attack while encrypting data by using ElGamal ECC?

I am using ElGamal ECC to encrypt my plain text data. I want to ensure that my data is safe from a Man-In-The-Middle attack. What methodology I can adopt to achieve this goal? How can we prevent a ...
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204 views

ECDH anonymous key exchange to avoid PKI

I want to use TLS to encrypt the communication between peers in a P2P network. Each peer has a well known 256bit peer identifier (the public key of a 256bit elliptic curve keypair). Both peers need ...
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297 views

ECC Complexity order of point addition, scalar point multiplication and selecting random point

I am facing this problem in calculating the order of a process which involves ECC point addition: $P+Q$ , scalar multiplication: $aP$, and selecting random points in the group. The group is of prime ...
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360 views

How do the following new (2013) ECC curves compare in security or efficiency? [closed]

I read about the following "safe" ECC curves and notably, secp256 and all the NIST curves are marked as "unsafe" when compared to more modern curves. I need a curve for signing or encryption, (or ...
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437 views

How do I convert the definition of E-521 into a curve definition a la Bouncy Castle?

I am currently trying to create an ECCCurve for E-521. Unfortunately, it is not currently a named curve in the library I am using, so I will have to define it manually. I am using the definition of ...
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260 views

Fast modular reduction

I am looking at ways to speed up modular reduction for the polynomial $$2^{256}-2^{32}-2^9-2^8-2^7-2^6-2^4-1$$ I have read the paper "Generalized Mersenne numbers" by J.A. Solinas, but it does not ...
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98 views

Where can I double check my elliptic curve results?

I am trying to do some elliptic curve calculations by hand, just to refresh myself on how the system works. I calculated some points and did some operations by hand. I am trying to double check my ...
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310 views

inverse problem about scalar multiplication on elliptic curve

Let $E$ be an elliptic curve over a finite field $F_p$. Given $n$ be a positive integer and $Q$ be a point on $E$, assume that $Q=nP$, how can we find this $P$? We can assume that $n|p-1$. If $n$ is ...
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197 views

EC equivalent for RSA-OAEP

I have some questions regarding aforementioned subject: Is there a EC equivalent of RSA-OAEP key transport/encryption algorithm ? Is ECIES-KEM sufficient ?
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150 views

Discrete log analog of ECM factoring algorithm?

Anecdotally, most factoring algorithms have a corresponding variant algorithm that can be used to attack the discrete log problem using similar ideas. Is there an analog of the elliptic curve (ECM) ...
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176 views

Generating non-supersingular elliptic curves for symmetric pairings

I am looking into the application of pairings in CPABE in particular. I've notice that the scheme uses a supersingular curve as the basis of the pairing. Looking through Ben Lynn's thesis for the ...
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308 views

Are there reference implementations of ECQV implicit certificates?

I am interested in exploring ECC implicit certificates, specifically using the ECQV protocol. While the actual implementation would not difficult to perform using building blocks provided by most ECC ...
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What happens if no final subtraction is done in Montgomery multiplication?

I'm doing Montgomery arithmetic modulo $N = 2^{255}-19$ for the Curve25519, picking $R = 2^{256}$ for Montgomery. After multiplying two numbers $0 <= A,B < N$ in the Montgomery representation ...
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34 views

Why recently Edward curve formulas being developed that deviates from unification?

Edward curves were considered initially because it provides a unified formula for both doubling and addition, thus having inherent side-channel resistance. But a lot of work has been done recently ...
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56 views

EC: Why does $h>200$ need to hold?

The class number of the principal order belonging to the endomorphism ring of E SHOULD be at least 200. From TR-03111 (pdf page 15). This value commonly is referred to as $h$. So my ...
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59 views

Which attacks are possible against raw/textbook elliptic curve?

A quick question, we know that raw RSA is a no go. To solve this we have different PKCS standards forcing structure on the input messages. For EC the story is something else. For signatures we have ...
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146 views

As a cryptographer, what are the things I should care about in my implementation of pairing functions?

As a beginner in cryptography, I do not know anything about different pairing types more than their names. So far, I know these names: Ate pairing, tate pairing, eta pairing, and r-ate pairing. I am ...
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110 views

Difference between “ECDH with cofactor key” and “ECDH without cofactor key”?

I need to use “ECDH with a cofactor key” for generating symmetric key. I have a fair idea on how ECDH works, but I don’t understand the cofactor part. What is the difference between ”ECDH with a ...
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273 views

Elliptic Curve based blind signature implementation

I want to use Elliptic Curve based blind signature scheme for my research. There is no proper implementation of ECC-based blind signatures. Can someone describe to me which things I need to follow ...
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74 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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172 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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119 views

Gallant-Lambert-Vanstone method

I am experimenting with the GLV method but cannot manage to get the right results according to the literature. I managed to find lambda, beta, split $K$ into $k_1$ and $k_2$ etc. for the curve I'm ...
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139 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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198 views

Is this ECC encryption key sharing method okay?

Is this encryption key sharing okay to use? Or is much better to use ECIES? $G$ = base point $a$ = Alice’s private key $b$ = Bob’s private key $A = aG$ = Alice’s public key $B = bG$ = Bob’s public ...
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Severity of Cooking NIST P Curve Constants

Bruce Schneier and Gregory Maxwell have both stated that they believe the constants chosen for NIST's P curves (i.e. P-256r) are cooked. DJB has put together a detailed list of red flags but, outside ...
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Which area of Maths should I pursue?

I would like to know which area of Mathematics would be most beneficial to cryptography. Surely Algebraic Number Theory and maybe to a lesser extend, Elliptic Curves, are closely linked to ...
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127 views

Simplified Example of ECC to use in the classroom

I have come up with the following rudimentary example of how ECC relates to asymmetric keys. Is this a valid explanation of ECC and its relationship to asymmetry? To only be deciphered by the person ...
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180 views

Elliptical curve cryptography key generation time

I am currently trying to learn more about Elliptical curve crypthography and have finally started to get things working and undestanding the different pieces. I've written a small project in C# and ...
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170 views

64 bit Elliptic Curve key?

For a simple proof of concept project i'm (attempting!) to do, i've started looking into openSSL elliptic curve cryptography. However instead of the standard key lengths, 160-512. I'm interested in ...
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94 views

Does a cofactor of an elliptic curve have to be an integer?

What are the implications of a curve having a non-integer co-factor for its generator point? Is that even possible?
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178 views

Is there a problem with this non-ECDSA message signing?

Is there a problem with signing a message like this? ...
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198 views

Elliptic curve brute forcing

I have elliptic curve of equation $y^2 \equiv x^3 -x $. And the coordinate of points $Q$ and $P$. I want to solve $Q=[k]P$ (where $k$ is the unknown) by testing all possible $k$. Is this the right ...
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107 views

Explanation of each of the parameters used in ECC

I'm having a very difficult time finding a clear explanation of the parameters used elliptic curve cryptography. I know for certain that $p$ is the number or order or whatever of the given field that ...
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138 views

Why does NaCL have different keys for signing and encryption?

I want to start using NaCL to sign messages that will go into a message queue, and I noticed that it generates different keys for each operation. Is there a reason for this? Can I not use the same PK ...
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120 views

Three-way key exchange with elliptic curves without pairing

Assume that there are three users, each with their own secret key $d_i$ and the corresponding public key $Q_i = d_i \cdot P$, such that $Q_i$ is a point on an elliptic curve and $P$ is a base point on ...
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323 views

Efficient algorithm for remainder calculation over prime field for ECC implementation?

I am working on 224-bit elliptic curve cryptography. In this 224-bit * 224-bit multiplication results 448-bit output. I am reducing 448-bit into prime field range( prime number $2^{224}-2^{96}+1$) ...
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439 views

File encryption with one keypair?

I'm working on a program that uses an ECC keypair in a (password protected) PKCS12 file (.pfx) to encrypt files. I like this method because I think it will be higher security (using ECDH to negotiate ...
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378 views

ECIES protocol - what does the || operation mean?

I am studying elliptic curves problems, which also includes study of related protocols such as ECIES. A there is a problem I don't understand operation $||$. What this operation mean? Some stuff is ...
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88 views

What aspect of elliptic curve encryption paradigms makes them especially susceptible to quantum based attack algorithms?

This was a statement made during a talk at today's DFN-CERT conference but unfortunately it wasn't explained further. Can anyone shed light on why elliptic curves are susceptible to quantum based ...
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205 views

Point decompression on an elliptic curve

I'm programming an elliptic curve cryptosystem and I'm having difficulty with decompressing points. The following information is from my project specification as to my understanding: Given a point ...
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122 views

What can be learned from the ciphertext of LibSodium's crypto_box_detached()?

LibSodium, (https://github.com/jedisct1/libsodium), has a function crypto_box_detached() which does authenticated encryption using the public key of the recipient ...
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111 views

Protection of Elliptic Curve Implementations against side-channel attacks [closed]

Recent fast elliptic curve implementations, for example a presentation at Eurocrypt 2014 (earlier presentation slides, the paper) talks about protection against only timing attacks. Why only timing ...
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150 views

Elliptic curve parameters

What's the meaning of 160 bit Curve in Elliptic curve ? or 192 or 224 or 256 and etc. And What is the standard for selecting this number of bit ? why they don't say 100 bit curve?
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291 views

With TLS and ECDHE, how does curve selection work?

Given that a TLS client and server have already agreed upon ECDHE for session key establishment, how does the selection of the actual elliptic curve ("domain parameters") being used for deriving the ...
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362 views

ECM Implementation is really slow

I followed the algorithms 14.4 (computes 1st and 3rd coordinates in (X,Y,Z)#k modulo n) and 14.5 (factorization using ECM) in David Bressoud's book 'Factorization and Primality Testing'. I think the ...
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48 views

Is it right about an example of bilinear map $e$?

The equation for the elliptic curve is $y^{2} = x^{3} + x$ and is defined over the field $F_{q}$ for some prime $q\equiv 3 \pmod 4$, and set $q=307$. Choose random generator $g=[182, 240]$. My ...
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50 views

Average/approximate difference in value between valid consecutive $x$ coordinates in ECC?

From my basic understanding not all values of $x$ coordinates can satisfy a given elliptic curve equation, i.e. some $x$ coordinate values are not valid points on the curve because $x^3+ax+b$ is not a ...
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112 views

Is only one shared secret generated by ECDH per key pair?

I'm confused about ECDH. Using their public keys and private keys, two entities can arrive on a shared secret. But from the equations I've seen, it looks like ONLY the numbers present in their key ...
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154 views

How does ECDHE_RSA key exchange mechanism work?

Using Wireshark, I found these data exchanged with google.com over TLS: Client Hello possible cipher suites and possible curve types (eg. secp256r1) sent Server Hello cipher suite selected ...
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In a additive group is it hard to calculate $bg$ given $ag, g, abg$

The ECDH problem defined that given $g,ag,bg$ it is difficult to calculate $abg$. But it is also difficult to calculate $bg$ given $ag,g,abg$. where $g$ is generator and a,b are elements of group.