Elliptic curves are a mathematical structure. In cryptography, it is common to use the structure $y^2 = x^3 + ax^2 + b$ over a finite field. Questions relating to elliptic curves and derived algorithms should use this tag and might also consider specific tags such as discrete-logarithm and ecdsa.

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What differences between Menezes–Vanstone ECC and ElGamal ECC?

After researching ECC encryption, I found that we can use ElGamal cryptosystem with elliptic curve and can we use Menezes-Vanstone cryptosystem with elliptic curve. What is the essential difference ...
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What are differences between $E(F_p)$ and $E(Z_p)$?

When I read some books about elliptic curve cryptography noticed that. sometimes symbolized elliptic curve over $F_p$ is $E(F_p)$ and sometimes symbolized elliptic curve over $Z_p$ is $E(Z_p)$. I ...
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Security benefits of Ed25519 generating signatures deterministically

I am reading on the Ed25519 curve, and I am trying to understand a claim. Here is the claim: Foolproof session keys. Signatures are generated deterministically; key generation consumes new ...
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279 views

ECC public key encryption and authentication - ECIES with ECDSA vs ECDH with AES

I'm currently working on a project where I want to establish a secure and authenticated communication channel between to entities, using Elliptic Curve Cryptography. Now I'm not really sure how to ...
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242 views

Are there any elliptic curve asymmetric encryption algorithms?

RSA offers the functionality of encrypting (short messages, or symmetric keys) with a public key, and decrypting with a private key. However, RSA key generation is extremely expensive, especially for ...
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Safe elliptic curve point addition using projective coordinates: How do I tell if the points are the same?

I am trying to implement elliptic curve point addition in hardware for NIST p256 and p384 curves. I have noticed the following issue with the suggested NIST routines: Routine 2.2.7 of ...
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Finite fields in elliptic curve

I have an elliptic curve defined over finite field where $S_1=aP$ . Is it valid to say that $S_1P$ can also be computed. $P$ is the generator of the group. What my real question is that. Should '$a$' ...
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145 views

How to derive formulas for addition and multiplication in Jacobian coordinates

Is there a way to derive the formulas for point addition and multiplication on elliptic curves in Jacobian format by yourself? How could I have derived these formulas by myself?
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Proving Non-Existence of ECC Backdoors

In light of the NIST Dual EC DRBG scandal, I was intrigued by a NIST slide (slide 9) that said the two points P and Q can be chosen so that the chooser can prove they don't have a backdoor. This ...
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140 views

Twisted curves in protocol

I've come to understand that twisted curves, as for instance defined in the Brainpool specifications, are $F(p)$-isomorphic to their regular $F(p)$ equivalents. So brainpoolP256r1 is isomorphic to ...
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126 views

How to handle the GCD(V,P) != 1 case when doing point addition or point doubling in elliptic curve cryptography

The equation for a finite field Elliptic Curve is of this form: $$y^2 \equiv x^3 + a * x + b \pmod{P}$$ When we do common EC operations like point doubling or point addition we need to calculate the ...
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107 views

Sextic twist optimization of BN pairing - cubic root extraction required?

I found the following paper really interesting: http://www.researchgate.net/publication/220378229_A_family_of_implementation-friendly_BN_elliptic_curves/file/79e4150b3a773beecd.pdf It allows ...
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394 views

Elliptic Curve Encryption Ciphertext Size

I'd like to know how much bigger is the ciphertext when encrypting a message using ECC encrytpion? ECIES (or ElGamal)
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369 views

Adding and multiplication in jacobian coordinates

How can I derive formulas for adding 2 points and multiplication by a scalar in Jacobian coordinates $(x,y) = (\frac{X}{Z^2},\frac{Y}{Z^3})$ over an elliptic curve?
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While generating a random Elliptic curve what are the conditions i have to considerd?

I want to generate a random elliptic curve over a prime field. What are the conditions I should satisfy? For the NIST recommended standard ECC-224 bit curve with prime $p=2^{224}-2^{96}+1$, a ...
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175 views

Asymmetric key derivation – Who derives the new pub key can't know the new private key

I'm looking for a peer reviewed protocol or algorithm that can do this, preferentially I need something based on elliptic-curves cryptography. Alice know the main public key of Bob (B-PUB-1). Alice ...
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150 views

Are RSA or ECC vulnerable to an attack where the same (unknown) plaintext is encrypted with multiple public keys?

I'm not sure what this attack model is called - it's not known-plaintext and also not quite cipher-text-only. It is similar to this question except the general case (not just two keys) and using keys ...
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157 views

Derive a public EC key from two public EC keys

Alice has two EC key pairs: $a_1$, $a_2$ are private keys (integers), $A_1$, $A_2$ are the corresponding public keys (points). Alice and Bob want to create a new public key $C$. Alice must prove that ...
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246 views

Elliptic curve parameter generation

I am curious of the details of how one would go about generating elliptic curve parameters. (I know standardized parameters exist, but I'm trying to understand both how they were generated and the ...
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259 views

Is there a field guide to ECC for the IT Security layman?

I'm trying to understand ECC from an IT layman's perspective and am trying to separate the theory from the standards, and understand why certain features are implemented or not implemented in the ...
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Is there a flaw in this ECC blind signature scheme?

Recently I've found the following work on the internet: An ECC-Based Blind Signature Scheme The paper claims to be an ECDSA blind signature however it seems that their scheme has a flaw in it. The ...
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Using the same private key for two ECC key pairs

Let $(d_1,Q_1)$ and $(d_2,Q_2)$ be ECC key pairs over two different elliptic curves (say NIST P-224 and NIST P-256). According to the Elliptic Curve Discrete Logarithm Problem (ECDLP), if the private ...
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Using single EC cert/keying material to derive symmetric encryption key (for storage)?

The situation involves a single party (single certificate) who would want to AES encrypt a file that they can later decrypt. Assume the EC certificate + EC keys have a purpose i.e. "File encryption" ...
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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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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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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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155 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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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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67 views

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

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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168 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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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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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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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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351 views

Why can ECC key sizes be smaller than RSA keys for similar security?

I understand how ECC is based on the discrete log problem and RSA on integer factorization. I've read several references that show how a solution to either of these problems can typically be adapted ...
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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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Can curve25519 keys be used with ed25519 keys?

Can curve25519 keys be used with ed25519? I'd prefer to use ed25519, but there isn't a fast java version. For my application, I'd like to use curve25519 until I can get a faster ed25519 for java. ...
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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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How are Elliptic Curve Cryptography and Pairing Based Cryptography related?

I have been doing a project that uses the PBC library developed by Ben Lynn. But I am still not clear on how PBC is related to ECC. I know that this is a site for complex crypto QA, but I did not know ...
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301 views

How to properly add ECDSA private keys?

I'm currently working on an application that requires me to add two ECDSA private keys in order to make a new private key. The result has to have the property, that its corresponding public key is the ...
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70 views

How many of primitive point on the elliptic curve?

In elliptic curves for cryptography, I know $nG=O$, where $G$ is a base point represented by $G=(x_g,y_g)\ on\ E(F_P)$, where $n$ is Order of point $G$. For example, $P(0,6)$ is a primitive point on ...
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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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219 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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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 ...
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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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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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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 ...
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293 views

How to represent point-at-infinity in affine coordinate

In projective coordinates point-at-infinity can be identified with z=0. How to identify the point-at-infinity in affine coordinate. Whether x=0 and y=0 can be considered as point-at-infinity in ...
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ECC algorithm pollard's $\rho$ complexity

One of the methods to break a ECDLP is Pollard's rho algorithm. When ECDLP is defined over a finite field $F_p$, and given a relation $S=w.T$, where S and T are a member of $F_p$. Then ECDLP is to ...
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What are these twist attacks with cost $2^{58.4}$ on NIST P-224 curve, and when do they apply?

This page on Twist security mentions a combined attack and a twist rho attack, applicable in particular to NIST P-224 curve with cost $2^{58.4}$ something, with no mention precise definition of ...