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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11
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2answers
983 views

Using same keypair for Diffie-Hellman and signing

Are there any security risks using a single key-pair for both key-exchange and signing? I'm mainly interested in using Curve25519 for key-exchange and Ed25519 for signing. But similar combinations, ...
62
votes
3answers
38k views

Why is elliptic curve cryptography not widely used, compared to RSA?

I recently ran across elliptic curve crypto-systems: An Introduction to the Theory of Elliptic Curves (Brown University) Elliptic Curve Cryptography (Wikipedia) Performance analysis of identity ...
11
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1answer
2k views

ElGamal with elliptic curves

I've searched some information on ECC, but so far I have only found Diffie-Hellman key-exchange implementations using ECC, but I don't want to exchange keys, I want to encrypt & decrypt data like ...
67
votes
3answers
21k views

Should we trust the NIST-recommended ECC parameters?

Recent articles in the media, based upon Snowden documents, have suggested that the NSA has actively tried to enable surveillance by embedding weaknesses in commercially-deployed technology -- ...
12
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1answer
658 views

Mapping points between elliptic curves and the integers

My primary question is: Is there an easy way to create a bijective mapping from points on an elliptic curve E (over a finite field) to the integers (desirably to $\mathbb{Z}^*_q$ where $q$ is the ...
6
votes
2answers
440 views

Besides key and ciphertext sizes what are other advantages of elliptic curve versions of various protocols?

There are elliptic curve variants of Diffie-Hellman, ElGamal, DSA and possibly other protocols/algorithms. I know that these elliptic curve variants have smaller key and ciphertext sizes which will ...
9
votes
3answers
426 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, ...
4
votes
2answers
526 views

Can we use elliptic curve cryptography in wireless sensors?

Can we use elliptic curve cryptography in wireless sensors? If so, how do you map points to message characters?
18
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4answers
11k views

Basic explanation of Elliptic Curve Cryptography?

I have been studying Elliptic Curve Cryptography as part of a course based on the book Cryptography and Network Security. The text for provides an excellent theoretical definition of the algorithm but ...
35
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6answers
13k views

Who uses Dual_EC_DRBG?

Recent news articles have suggested that the NSA may be involved in trying to influence the cryptography in public standards or commercially deployed software, to enable the NSA to decrypt the ...
5
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0answers
671 views

Elliptic curve cryptography related key attacks

This question is an extension of Families of public/private keys in elliptic curve cryptography As described above, bitcoin "type 2" deterministic wallets use a root private/public key pair, where ...
6
votes
1answer
485 views

Why are elliptic curve variants of RSA “chiefly of academic interest”?

Yesterday I was thinking about elliptic curve variants of popular protocols/algorithms (ECDH, ECES[1], etc) and the thought occured that I had never seen an elliptic curve variant of RSA. My ...
3
votes
1answer
410 views

ECDSA vs RSA: Performance on Android platform and surprising results

For our privacy-preserving protocol, an encrypted channel is established. In order to protect our system from man-in-the-middle attacks, signature-based approach is used. After we've implemented it ...
3
votes
4answers
3k views

How does one calculate the scalar multiplication on elliptic curves?

I found this example online: In the elliptic curve group defined by $$y^2 = x^3 + 9x + 17 \quad \text{over } \mathbb{F}_{23},$$ what is the discrete logarithm $k$ of $Q = (4,5)$ to the base ...
2
votes
2answers
208 views

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 ...
1
vote
0answers
45 views

Smart card Strong authentication / Verification ( fingerprints)

I'm trying to make a strong authentication software and embedded software in a java card. I have found many papers and publications about the subject… too much information to process and I'm working ...
0
votes
1answer
61 views

How to calculate kinv from the given k value

I am implementing an ECDSA NIST test vectors verification application. The test vectors are taken from http://csrc.nist.gov/groups/STM/cavp/#09. One of the test vectors is given below: ...
20
votes
3answers
19k views

How can I use SSL/TLS with Perfect Forward Secrecy?

I'm new to the field of cryptography, but I want to make the web a better web by setting up the sites that I host with Perfect Forward Secrecy. I have a list of questions regarding the setup of ...
29
votes
1answer
25k views

Explaining weakness of Dual EC DRBG to wider audience?

I have an audience of senior (non-technical) executives and senior technical people who are taking the backdoor in Dual_EC_DRBG and considering it as a weakness of Elliptic curves in general. I can ...
7
votes
2answers
492 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? ...
14
votes
3answers
1k views

Can ECDSA signatures be safely made “deterministic”?

Using the terminology of the ECDSA wikipedia page, ECDSA (and DSA) signatures require a random k value for each signature which ensures that the signature is different each time even if the message ...
6
votes
2answers
2k views

With OpenSSL and ECDHE, how to show the actual curve being used?

Using openssl s_client -host myserver.net -port 443 I can see the cipher negotiated is indeed using ECDHE for session key ...
3
votes
1answer
371 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 ...
13
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2answers
1k views

Does the elliptic curve (EC) cryptosystem outperform RSA and DL cryptosystems?

Throughout the literature, it is stated that EC cryptosystems outperform RSA and Discrete logarithm cryptosystems, but I cannot understand how ECC would be more efficient than RSA and DL in terms of ...
10
votes
2answers
916 views

How does the MOV attack work?

What exactly is the MOV attack, how does it actually work, and what is it used for? It's explained briefly here and I'd like to know what it is more / what is it fully used for.
8
votes
1answer
478 views

Compressing EC private keys

For reasonable security, EC private keys are typically 256-bits. Shorter EC private keys are not sufficiently secure. However, shorter symmetric keys (128-bits, for example) are comparably secure. I ...
7
votes
2answers
703 views

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 ...
4
votes
1answer
366 views

Trying to better understand the failure of the Index Calculus for ECDLP

So I'm going to give you guys my understanding and then if you would be so kind as to tell me where I'm off the mark (hopefully I'm not completely wrong). So basically the index calculus for the ...
4
votes
1answer
451 views

Families of public/private keys in elliptic curve cryptography

I'm looking for a related key scheme for elliptic curve cryptography. The basic idea would be that there would be a master public key and a master private key. From the master public key, you could ...
3
votes
1answer
365 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)
6
votes
1answer
128 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 ...
5
votes
1answer
109 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[] ...
4
votes
3answers
263 views

Why are elliptic curves better than cyclic groups?

The set of points of an elliptic curve over a finite field is isomorphic to the direct product of two cyclic groups (i.e. $E(F_{p^n}) \cong Z_{s} \times Z_{t})$. What is the advantage of representing ...
3
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0answers
119 views

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 ...
3
votes
1answer
324 views

Subtracting a point in elliptic curve cryptography?

I've had lots of practice adding points for my crypto class. However I've run into a situation where I need to subtract two points for decryption: ...
2
votes
1answer
139 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 ...
2
votes
1answer
116 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: ...
2
votes
1answer
68 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
votes
3answers
251 views

understanding pairing $e:G \times G \to G_T$ and ( Decision)BDH assumption

From DrLecter's comment, I know that DDH problem can be efficiently solved with this $$e(g^a,g^b)\stackrel{?}{=} e(g,g^z).$$ I have some trouble to understand this map $e:G \times G \to G_T$. Am I ...
2
votes
0answers
207 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 ...
1
vote
1answer
135 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 ...
1
vote
2answers
129 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 ...
1
vote
1answer
148 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 ...