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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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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counting points not on elliptic curve

Given an curve with equation $y^2=x^3+ax+b$, I want to find the number of pairs $(a,b)\in \mathbb{F}_p \times \mathbb{F_p}$ NOT on the curve. How do I do it? I have an intuition that it is $p$, but ...
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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 ...
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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 ...
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137 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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$\phi$ 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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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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151 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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378 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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230 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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258 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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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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232 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 ...
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About Elliptic Curve ElGamal, 3 simple problems I have trouble with

In Elliptic Curve ElGamal, why are a=b=1 always legal for primes whose lengths are no shorter than 11(2) bits long? Is there any reason why the Point at Infinity can always be encoded as (0,0)? ...
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1answer
240 views

Signature algorithm SHA 1-2 with ECDSA

Can someone please explain what key sizes are required for the ECDSA algorithm? I tried a 128 bit EC Key for SHA1withECDSA and it throws an error. However with 256 bit key I could run the algorithm. ...
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What is so special about elliptic curves?

There seems to be sources like this, this also, and some introductions that discuss elliptic curves in general and how they're used. But what I'd like to know is why these particular curves are so ...
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121 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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147 views

Why is a simple hash into $G_2$ for (certain) pairing based crypto not possible?

In the paper Pairings for Cryptographers we read about what the authors call a type 2 pairing in which we have a "pairing friendly curve $E$ over $\mathbb{F}_q$ with embedding degree $k>1$ and ...
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241 views

openSSL ECDH private key size

When you are using a named curve like P-256 in openSSL, is there any standard key size for ECDH private key keys? If you look at the ec_key.c file in the openSSL ...
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1answer
217 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: ...
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125 views

With TLS and ECDHE, how does curve selection work?

Given that a TLS client and server have already agreed upon ECDHE for session key establishment, how does the selection of the actual elliptic curve ("domain parameters") being used for deriving the ...
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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 ...
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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, ...
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51 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 ...
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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 ...
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107 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 ...
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1answer
150 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 ...
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1answer
198 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 ...
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Using same keypairs for crypt and sign with elliptic curve [duplicate]

I asked my question on a mailing list ( http://www.metzdowd.com/pipermail/cryptography/2013-October/018061.html ) and i received many answers, which are confused me. So i try to ask here, it is ...
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2answers
170 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 ...
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137 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 ...
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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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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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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 ...
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123 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 ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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How can someone predict the bits of a not-so-PRNG like Dual_EC_DRGB?

Given the recent revelations of the NSA engineering a backdoor into the Dual_EC_DRBG algorithm - I'm curious, exactly how, given the number of variables involved in generating the random bits could ...
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1answer
336 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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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 -- ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...