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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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 ...
5
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2answers
122 views

Is it safe to reuse ECDH asymmetric keys for authentication?

Alice, Bob, and Carol each generate ECDH keypairs. Alice and Bob establish a communication channel and negotiate an AliceBob secret. The question is: Is it safe for Alice and/or Bob to reuse their ...
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1answer
103 views

How can I find the order of the group that an elliptic curve is defined over?

I have a Weierstrass elliptic curve ($y^2=x^3+a \times x+b \mod p $) How can I find the order of the group itself? I have seen Mathematica has a GroupOrder[] ...
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1answer
129 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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1answer
180 views

Which of following new (2013) ECC curves is the most secure or efficient? [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 ...
2
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1answer
96 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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3answers
374 views

Elliptic Curves of different forms

Looking at http://safecurves.cr.yp.to/ to find a safe curve, I find that most curves described here are of a different form from that generally used. In Bouncy Caslte, for example, ...
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258 views

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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1answer
102 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 ...
2
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1answer
267 views

Mapping of message onto elliptic curve and reverse it

I would like to perform a variant of Elliptic Curve ElGamal in java using the BouncyCastle libraries. I currently face the difficulty of mapping a message $m$ onto the elliptic curve $E_p$. I have so ...
3
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1answer
127 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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1answer
106 views

cryptographically good random elliptic curves?

After an answer here, about generate elliptic curves, I've start thinking about the algorithm. The mentioned algorithm will produce curves in the Weierstraß Reduced Form (WRF) over finite fields: ...
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2answers
69 views

Finding an x such that xP = (11,44) on an elliptic curve

Given the elliptic curve $$E:y^2 = x^3+17x+5 \mod 59$$ with point $P = (4,14)$, how do I find $x$ such that compute $x\cdot P = (11,44)$ Is there a mathematical method to compute $x$, or do I ...
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2answers
121 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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0answers
79 views

Weil pairing implementation - low level programming language

I'm just started to studying elliptic curve cryptosystems. My one month-goal is to write a simple signature system based on the Weil-pairing. Some parts of it can be written without deeper ...
2
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1answer
129 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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0answers
57 views

Sextic twist of BN pairing parameters vs security

I've previously asked questions on BN pairing parameters. Here's one more. In the BN construction, one is working in a subgroup of a curve over an extension field $\mathbf{F}_{p^{12}}$ for some ...
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0answers
88 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 ...
3
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1answer
99 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 ...
2
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2answers
468 views

Using ECDSA keys for encryption

I know that ECDSA is used for signature only, but I wonder if I can use the public/private Elliptic Curve keys for encryption too. I have ECDSA SSH public keys and I wonder if I can use them to ...
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1answer
81 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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67 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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0answers
93 views

Safe and computationally efficient way to verify a curve25519 identity?

A client identifies itself as a curve25519 public key. The server wants to verify the client owns the associated private key. Is there a safe and computationally efficient way of doing so? Which ...
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1answer
93 views

If its possible to derive the public key from a private key, why can't we go in reverse?

I'm looking at source code for BitcoinJ that derives a public key from the private key. ...
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1answer
126 views

ElGamal with elliptic curves I

It is very interesting to see @tylo's answer on ElGamal with elliptic curves. Instead of mapping the message to the elliptic curve point it just reduces an elliptic curve point to its $x$ coodrinate. ...
3
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1answer
166 views

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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1answer
129 views

What are some restrictions when converting Montgomery Curves into Weierstrass Curves?

I want to represent a Montgomery Curve (curve25519) in Weierstrass form as a personal exercise. After doing some math and referencing the conversion equation at ...
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1answer
282 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 ...
2
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3answers
145 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 ...
2
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1answer
274 views

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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0answers
117 views

inverse of scalar multiplier in ECC

I am learning to use ECC. i got into situation where i have $Q=abG$, where $G$ is the generator of the finite field formed on EC using a prime $p$ modulus and $a$ , $b$ are random numbers. now suppose ...
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1answer
116 views

Are there groups where the computational Diffie Hellman problem is easy but the discrete log problem is hard?

I know that there are elliptic curve groups, used in pairing-based cryptography, where the decisional Diffie Hellman problem (ie. given $g$, $g^a$, $g^b$ and $c$, determine if $c = g^{ab}$ is easy but ...
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2answers
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Is there a feasible method by which NIST ECC curves over prime fields could be intentionally rigged?

The NIST elliptic curves P-192, P-224, P-256, P-384, and P-521, prescribed in FIPS 186-4 appendix D.1.2, are generated according to a well defined process, but using an arbitrary random-looking seed ...
5
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1answer
1k views

ECDSA vs ECIES vs ECDH

Recently I started studying ECC and I just loved it. I want to transfer some big data (like 3KB), What is the best method, ECDSA, ECIES, or ECDH (and why)? I am confused, how should I choose between ...
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1answer
342 views

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
463 views

Is Curve25519-java secure?

I have only about 2 weeks of cryptography experience mostly in the form of questions on bitcoin.se. Is Curve25519-java up to date with current Curve25519 standards? Is Curve25519 itself secure? ...
4
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1answer
185 views

EC ElGamal versus static+ephemeral ECDH

A client application needs to encrypt a UDP datagram for a server with known EC public key $P$. Performing a full ECDH key exchange would defeat the benefit of using UDP as a connectionless protocol. ...
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1answer
107 views

Elliptic Curve Verifiable Secret Sharing

I'm reading this paper, which on page 3(Section IV.C) presents a Jointly Random Verifiable Secret Sharing Scheme for Elliptic curves. The algorithm makes sense to me save for this part: "Each ...
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111 views

Where can I find examples of ECC implemented in VHDL?

I am looking into implementing ECDSA signature and ECDH key agreement in a Xilinx FPGA. All the examples I have found of VHDL implementations skip over how to construct the low level ECC primitives ...
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1answer
95 views

Is an elliptic curve over $\mathbb{F}_p$ order preserving for the points $(x,y) \in \mathbb{Z}_p$?

Let $E\colon y^2=x^3+ax+b$ be an elliptic curve, and consider its realisation over the finite field of prime order $p$: $E(\mathbb{F}_{p})$. Then if $0<a,b$ is the following true? $$\forall ...
2
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1answer
145 views

counting points on elliptic curve

Given an elliptic curve with equation $y^2=x^3+ax+b$, and i want to find the number of points $(a,b)\in E(\mathbb{F}_p)$ where the polynomial has repeated roots, how do i do it? I have an intuition it ...
2
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2answers
191 views

Graphically representing points on Elliptic Curve over finite field

I have taken elliptic curve $E\colon y^2=x^3-4x+20$, defined over $\mathbb{F}_{29}$. The number of points on the curve, $\left|E(\mathbb{F}_{29})\right|=37$. I took base point $P=(1,5)$, and got ...
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0answers
91 views

$\phi$ function in Dual_EC_DRBG

I am trying to understand the operation of the Dual_EC_DRBG. I'm reading the formal specification (SP 800-90) and can't seem to find a definition of the $\phi$ function used throughout the definition ...
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82 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 ...
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1answer
162 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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446 views

When using Curve25519, why does the private key always have a fixed bit at 2^254?

When using Curve25519, the private key always seems to have a fixed bit set at position 2^254. Why is that? Is there any good reason to use a fixed positioned most-significant-bit in the private key? ...
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2answers
271 views

How can I implement the elliptic curve MOV attack myself?

I understand and have implemented elliptic curve signatures in Python without the use of libraries like Sage, and would like to implement the MOV attack against certain weak types of elliptic curves. ...
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1answer
327 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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2answers
176 views

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 + ...
5
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1answer
270 views

In elliptic curve cryptography, how is “A dot A” computed?

I was just reading Ars Technica's primer on ECC. Somewhere near the middle of the second page, the author introduces the "dot" operation that takes an elliptic curve and two other known points, giving ...