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.

learn more… | top users | synonyms (1)

7
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
3answers
495 views

Timing Attacks on ECDSA, ECDHE, AES and SHA2

Are there any known timing attacks (both practical and theoretical) on any implementations of the following? ECDSA (I'm aware of this one - are there any applicable to prime fields?), ECDHE (again, ...
2
votes
1answer
87 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 ...
3
votes
1answer
178 views

Can keys from Bitcoin's Hierarchical Deterministic Wallets be correlated (reducing privacy)?

I'm trying to understand if the feature "Hierarchical Deterministic Wallets" in Bitcoin allows for complete privacy of all derived keys, and if any of those keys can be associated with each other ...
1
vote
1answer
118 views

Efficiency of computing $e(P,Q)$ Vs $g^a \pmod{p}$?

The definition of $e$ can be seen here. I want to know the accurate comparison of efficiency between $e(\cdot,\cdot)$ and $g^a \pmod{p}$. If computing $e(P,Q)$ is less efficient than computing $g^a ...
1
vote
1answer
243 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 ...
4
votes
1answer
361 views

Blockwise Montgomery multiplication

I have to implement a 256*256 bit Montgomery multiplier for pairing computations. The straightforward approach is to use a bit-serial version, but I would like to utilize the built-in 64*64 bits ...
0
votes
2answers
322 views

Reuse of a DH / ECDH public key

I was wondering whether it is safe to use the same DH or ECDH key pair in more than one key agreement, particularly if these public keys are in a public registry. These public keys could be used by ...
0
votes
1answer
265 views

Key space: Dense and sparse

I'm taking a cryptography class and I come across these terms dense and sparse key space allt he time. What do they mean? As far as I can I understand, dense key space means that there are more ...
2
votes
1answer
335 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 ...
3
votes
1answer
164 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 ...
1
vote
0answers
237 views

Does BouncyCastle (for ECC) resist timing attacks?

I need to extend the TLS protocol to be able to use other key exchange scheme based on elliptic curves. I am planning to use BouncyCastle's implementation in Java and in .NET. I am worring about ...
1
vote
1answer
188 views

Curve parameter for hyperelliptic curve cryptography?

RFC5639 defines some curve parameter for Elliptic curve cryptography. Aren't there any curve parameter database for Hyperelliptic curve cryptography? What I can only find was that written in this ...
2
votes
1answer
246 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" ...
2
votes
0answers
460 views

Understanding elliptic curve encryption

I'm having a hard time understanding the elliptic curve encryption. One thing thing I don't understand is listing all the points on the curve (mod p). Suppose I have the following elliptic curve: y^2 ...
3
votes
3answers
269 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 ...
31
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 ...
6
votes
1answer
249 views

What exactly could be accomplished with a backdoor in Dual_EC_DRBG?

Assume that some entity really holds the private key corresponding to the recommended/dubious constants of Dual_EC_DRBG. According to this presentation, they would be able to reconstruct the internal ...
3
votes
1answer
842 views

Possible ECC backdoor and its impact on Internet traffic

In a recent article, Bruce Schneier suggested that he prefers classic discrete log crypto over ECC because "I no longer trust the constants. I believe the NSA has manipulated them through their ...
2
votes
1answer
407 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 ...
75
votes
3answers
24k 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 -- ...
4
votes
2answers
298 views

Proof of elliptic curve difficulty

Are there any proofs that cryptographic functions on an elliptic curve are any more difficult than the analogues over modulo arithmetic? While at present, ECC appears to be more difficult, as it is ...
36
votes
6answers
14k 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 ...
2
votes
1answer
96 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 ...
3
votes
2answers
313 views

Elliptic Curve Cryptography Encryption Results

I'm playing around with a package that does Elliptic Curve Cryptography from http://jecc.sourceforge.net/ Every time I encrypt a value it produced a different result (same private key). However I'm ...
3
votes
2answers
342 views

Finite fields and ECC

I understand modular arithmetic(or at least I think I do!) and I've tried to read and learn about how the Math in RSA works(and I think it went pretty well). I've been reading up on ECC and it looks ...
13
votes
1answer
3k 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 ...
-1
votes
1answer
72 views

Is size Q equal to size SHA(Q)? [closed]

Assume d is a 128 bit random integer and P is base point of an elliptic curve and Q = dP is a point on the elliptic curve and SHA is a hash function with 128 bit output, my question is: Is size Q ...
1
vote
1answer
119 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 ...
4
votes
0answers
549 views

Reasons for Chinese SM2 Digital Signature Algorithm

In the IETF RFC draft named "SM2 Digital Signature Algorithm" a signature algorithm is specified. The RFC does however not mention why this signature algorithm has been defined. Nor does it specify ...
8
votes
2answers
2k views

Why is the P-521 elliptic curve not in Suite B if AES-256 is?

In the NSA's document, "The Case for Elliptic Curve Cryptography", we have ...
6
votes
6answers
840 views

Generate Elliptic Curve Private Key from User Passphrase?

I'd like to generate a private elliptic curve key from user input like pass phrase. Is the best way to do this with a key derivation function like PBKDF2? Is there a better way? Edit (based upon ...
1
vote
1answer
204 views

Relationship between Elliptic Curve Discrete Log, Integer Discrete Log, and Integer Factorization

I am trying to look into a relation between the following three problems which are widely used to build public crypto systems: Integer Discrete log Elliptic Curve Discrete log Integer Factorization ...
2
votes
1answer
134 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 ...
4
votes
1answer
346 views

How Were secp*k1 elliptic curve generators chosen?

The Koblitz elliptic curves specified in the SEC2 document https://docs.google.com/viewer?url=http%3A%2F%2Fwww.secg.org%2Fcollateral%2Fsec2_final.pdf all have the nice feature that the parameters are ...
3
votes
2answers
150 views

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 '$a$' ...
6
votes
1answer
815 views

Why Elliptic curve cryptography are not popular in practice

RSA and ElGamal can be implemented using the technique of Elliptic curves. I am confused on why the it seems that Elliptic curves are not so popular in cryptographic applications since they provide ...
5
votes
2answers
743 views

Converting Ed25519 public key to a Curve25519 public key

I understand that: $$x_{montgomery} = \frac{1 + y_{edwards}}{1 - y_{edwards}}$$ Using the libsodium ed25519 implementation, I have tried to write the following: ...
4
votes
1answer
205 views

How can I tell which curve a given ECDSA implementation uses? (P-521 or something else)

I'd like to test and see if certain software uses P-521 ECC curves, or if it uses another variant. Without having access to the sourcecode, or the specification, is there any way for me to test which ...
4
votes
3answers
145 views

using elliptic curve point multiplication as a key stretching method

My friend came up with the following idea: assuming we agreed on curve parameters, use the following algorithm for key stretching/derivation from user-entered password. Pad the ascii representation ...
1
vote
3answers
208 views

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 ...
1
vote
2answers
170 views

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

Is there a problem with signing a message like this? ...
2
votes
0answers
183 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 ...
3
votes
2answers
207 views

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" ...
20
votes
3answers
24k 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 ...
2
votes
0answers
261 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 ...
3
votes
1answer
295 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 ...
4
votes
3answers
329 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 ...
1
vote
1answer
418 views

Smart Card Basics

I want to implement some of the basic encryption algorithms on smart card, could any body guide me how to program a smart card, which tools (hardware and software) I should have, and if these tools ...
1
vote
1answer
257 views

ECC Point Multiplication of Product

I can calculate $Q = a\,b\,G$ in several ways: $Q = a \, (b \, G)$ or $Q = b \, (a \, G)$. These give the same result, as expected. But if I do $c = (a \, b) \bmod n$ where $a \, b$ is much greater ...