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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Which eliptic curves in OpenSSL 1.0.1f meet all / most of the SafeCurves requirements? [closed]

I am using nginx compiled with OpenSSL 1.0.1f (most current release available). Nginx allows administrators to set a configuration parameter called ssl_ecdh_curve, ...
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3answers
139 views

Is there any alternative for extended euclidean algorithm to perform modulo division?

I'm implementing point addition and point doubling operations for elliptic curve cryptography. I have implemented extended euclidean algorithm to perform modulo division. It appears the that ...
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327 views

Measure ECC key size

I have implemented a ECC key generation scheme successfully. Now I need to find ECC key sizes of each generating key pairs. I assumed that ECC key size is the size of the ECC private Key. So I would ...
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2answers
358 views

Why are there only positive value points on an elliptic curve?

I read about elliptic curve cryptography $E$ over $Z_p$ where $p$ is prime number and $G$ is a base point on the curve. I noticed the points resulting from multiplication e.g. $2G$,$3G$,.....,$(N-1)G$ ...
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330 views

Is there a 1:1 mapping between private and public EC keys?

After asking: Are all possible EC private keys valid? I learned that all 32 byte (256 bit) values greater than 0 and less than n are all valid private keys. This means that 99% of all 256 bit values ...
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What curve and key length to use in ECDSA with BouncyCastle

I'm developing a client/server system in Java which is not interacting with third party software, so I don't have to worry about compatibility. At a certain point, I need the client and server to ...
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1answer
327 views

Elliptic Curve Factorization: Why are elliptic curves suited for this kind of task?

Currently I'm working on a presentation of a paper that talks about the factorization of large numbers. In the paper elliptic curves are presented as a way to factorize large numbers. After hearing a ...
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484 views

Hide a weakness in ECC by choosing the prime or one of the curve coefficients

Suppose you are given a value $c$. Can you find a prime $p$ and an integer $b$ such that the elliptic curve $$E: y^2 \equiv x^3 -3x + b \pmod p$$ is cryptographically weak? You need to choose ...
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273 views

Adding points on Elliptic Curves

How do we add the integer points $P=(-1, 4)$ and $Q=(2, 5)$ on the elliptic curve of the form $y^2=x^3+17$ ?
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206 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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431 views

Is the term “Elliptic Curve Discrete Logarithm Problem” a misnomer?

I have just started studying Elliptic Curve Cryptography, and I have this doubt. In ECC the group operation is addition (and not multiplication). So, why is ECDLP stated as a variation of the discrete ...
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251 views

What is the fastest elliptic curve operation f(P) in affine coordinates such that f^n(P)=P only if n is large?

I'm working with the affine representations of points of the Secp256k1 elliptic curve (from Bitcoin). I've read many papers that show that computing some functions, like $f(P)=3P$ can be computed ...
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339 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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1answer
298 views

ECC partially blind signature scheme verification

Continued from Is there a flaw in this ECC blind signature scheme? The problem I needed a partially blind signature scheme for one of my projects, but couldn't find one on the internet, so I've made ...
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2answers
241 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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363 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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349 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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1answer
166 views

What is the difference between order of base point and curve order in EC? [duplicate]

When I was read about the elliptic curve cryptography I found some definition about domain parameter of elliptic curve like the follow. But I did not understand something $p$: prime number. $a, b$: ...
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1answer
631 views

Why is 224 bit ecdsa faster than 192 bit ecdsa?

I ran several benchmarks using openssl on 2 different computers and I got a surprising result. for the Nist 192 bit curve the benchmark result is ...
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3answers
487 views

Can a EC private key be derived from a public key?

I understand that the public key does not expose the private key. That is not the question. The question is: Given a EC public key, can a different, but plausible and functional private key be ...
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1answer
206 views

Is either brainpoolP320r1 or brainpoolP320t1 a SafeCurve?

Not all elliptic curves are safe to use for cryptography, especially from an ECC safety perspective. The site http://safecurves.cr.yp.to/index.html shows that two tested Brainpool curves, ...
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1answer
174 views

Verifying multiplicative inverse on a prime field in NIST's ECDSA_Prime.pdf

I am trying to learn about the Elliptic Curve Digital Signature Algorithm (ECDSA) by verifying the results in some example calculations. I found a PDF of example ECDSA calculations from NIST here: ...
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290 views

Generating bilinear pairing parameters - running time of finding member of p-torsion group

Update: Question completely rephrased. I want to create the parameters for a bilinear pairing (the Tate pairing in this case). In case you're interested I'm following this thesis, specifically the ...
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1answer
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ECC - Point Addition/Point Multiplication

So I have a very beginner-esque knowledge of ECDSA and I'm trying to write something in python to take a private key and output the public key (Basically from what I understand just trying to do the ...
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1answer
330 views

Brute force attack expected running time

I am a bit confused about the expected running times of brute force attacks on different cryptosystems. So let's assume a key size of $2^n$ bits. Symmetric key cryptography: $E(brute)$ = ...
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Scalar Multiplication on Elliptic Curves

In the elliptic curve: $y^2 = x^3 + 20x + 13 \bmod{2111}$. Using the point $P=(3, 10)$ I am wondering how to multiply this point by the scalar $57$? I realize I can write $57*P$ as $2^5*P + 2^4*P + ...
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1answer
394 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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1answer
84 views

Is there an asymmetric algorithm that can perform double encryption?

I'm looking for any asymmetric algorithm can perform the serial encryption, by serial I mean double or more encryption with different key(same key size). RSA can not do it since the cipher of the ...
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1answer
75 views

How to perfrom modular division while numerator is lesser than the denominator?

I'm implementing point addition and point doubling of elliptic curve cryptography. The formula that I'm using for slope is Point addition: $S = \frac{(P_y-Q_y)}{(P_x-Q_x)}$ where $P$ and $Q$ are the ...
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2answers
30 views

With EC secp256k1 is there a way of transforming a function of the private key to a function of the public key?

A key pair has a private key $D_A$ and a public key $Q_A$. $D_A$ is an integer less than the curve's $n$. Is there any (boolean) function of the private key $f(D_A)$ which can be transformed into a ...
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1answer
112 views

Can I use an ECDH Shared Secret from the same Private / Public Key Pair?

I'd like to know if using the ECDH shared secret of a static EC Private Key with it's own corresponding static EC Public Key causes a problem / weakness. (edit) not asking if it's ok to re-use the ...
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1answer
70 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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1answer
114 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 ...
2
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1answer
111 views

Roots in modulo field

I have a point $(X,Y)$ on an elliptical curve $E(a,b)$ where $a=-3$ and $B$ is a large number that is in hexadecimal from -51BD. To compress this point oficially in a program, we know that every $X$ ...
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123 views

What is primary security in ECC?

I read the following paragraph in a book about Elliptic Curve Cryptography, but didn't understand it: The primary security in ECC is the parameter $n$; where $n$ is Order of point $G$, that is $n$ ...
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1answer
919 views

Why does openssl use SHA1 in ECC when I use secp384r1 curve

I need a small clarification that why openssl using SHA1 in ECC when I am using secp384r1 curve, but in rfc they are saying we should use SHA2. Thing here is am using nanoECC in my DTLS, nanoECC ...
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1answer
277 views

How is the curve equation used in ECC?

I have a hard time learning exactly how the elliptic curve equation is used in the ECC. $$y^2 = x^3+ax+b$$ If someone knows and could explain to me in simple steps how this is done or a link to it ...
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1answer
122 views

How do I know if a given curve requires a FpCurve F2mCurve or ECCCurve?

I'm trying to read a public key into Bouncy Castle (secp256k1) and need to choose from the following objects FpPoint; FpCurve; or ...
2
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1answer
316 views

Can Secp 256 K1 curves “map” to a value on FIPS 186-3 or P-256?

I'm looking at Secp 256K1 vs UProve's FIPS 186-3 or P-256 implementation. Is there any relationship between the curves such that I can consistently "map" or "project" values from one curve to ...
2
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1answer
427 views

Is using Ed25519 parameters in ECDSA safe?

I recently discovered the Curve25519 key exchange lib and the Ed25519 signature lib. Due to the speculations about NIST-designed curves, there is a chance that I ditch them and use the curves above ...
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1answer
282 views

Elliptic curve cryptography attack vector

I would expect a complicated answer for what seems like a simple question about Elliptic curve cryptography. I've read several entries here such as "Elliptic curve cryptography related key attacks" ...
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1answer
165 views

Verify Messages to Embedded Device

I'm building an embedded device that I plan on distributing. Periodically the device will poll my server to check for updates and commands. I'd like the device to verify that any messages(JSON strings ...
2
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1answer
2k views

BouncyCastle Elliptic Curve implementation

I'm implementing ECDH key exchange in C# using the BouncyCastle library and I'm having a hard time understanding the elliptic curve side (FpCurve). ...
2
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1answer
94 views

Elliptic curve ElGamal with homomorphic mapping

I am interested in ElGamal due to the fact that you can achieve some degree of homomorphic properties. I became interested in applying ElGamal to elliptic curves, and found this other question with an ...
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1answer
49 views

Are private and public key sizes of Elliptic curve related?

I'm new to elliptic curve cryptography. I just want to know in the case where I take a random number (private key) and find its associated public key, does the size of the public key depends upon ...
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1answer
37 views

Is there a single-use signature scheme, where a second use of the private key discloses it to the world?

With ECDSA (and possibly DSA too) I'm aware that if the same value for $k$ is used with the same private key $D_A$ to sign two different messages, then anyone possessing the two messages $m_0$ and ...
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1answer
271 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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2answers
259 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 ...
2
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1answer
214 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 ...
2
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1answer
178 views

Point addition in NaCl/libsodium (Curve25519)

In NaCl and libsodium, the crypto_scalarmult function implements the operation $Q = kP$ (scalar/point multiplication). There doesn't seem to be a function for point ...