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

Is there a problem with signing a message like this? ...
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141 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 ...
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190 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" ...
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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 ...
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216 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 ...
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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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279 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 ...
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343 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 ...
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218 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 ...
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192 views

Can somebody explain the major contributions of the tenants of the Gödel Prize 2013?

As you may know, the Gödel Prize 2013 will be awarded this year to cryptographers (see this ACM press release). The people awarded are Antoine Joux, the team of Dan Boneh and Matthew K. Franklin. Can ...
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127 views

Complex Numbers on Elliptic Curves & Usage in Tate Pairing

I'm working with understanding the internals of the Tate Pairing. I was going through an example of the curve $E: y^2 = x^3 + 3x$ over $\mathbb{F_{11}}$. The author is showing the computation of ...
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207 views

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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calculating beta for elgamal elliptic curves [duplicate]

Suppose we use elgamal elliptic curves for secure communication. Bob selects a prime $p$, an elliptic curve $E$, a point $\alpha$ on $E \pmod p$, and a secret integer $f$. Suppose that Bob has ...
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383 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 ...
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1answer
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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158 views

Elliptic Curve is DH function or PKI?

can we reuse same ECC key on TLS for long terms or it must be used just once? (i mean can we use ECC like RSA?) is there patent free ECC implementation ?
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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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410 views

The utility of elliptic curve cryptography

Suppose that the only public key cryptography schemes that we knew were Diffie Hellman, RSA and ElGamal. How much would this set civilization back? Are there important applications of elliptic curve ...
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180 views

EC equivalent for RSA-OAEP

I have some questions regarding aforementioned subject: Is there a EC equivalent of RSA-OAEP key transport/encryption algorithm ? Is ECIES-KEM sufficient ?
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132 views

How fast can a wrong decryption key be detected using ECC?

When can a decryption function detect that the ECC key I use for decryption is incorrect? Is it possible to do that during initialization, or does the complete message have to be decrypted to do that? ...
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2answers
304 views

How do I unpack the x and y values from the BITSTRING in a DER ECDSA public key?

In ASN.1, the X and Y values for a 256-bit elliptic curve key are stored as a single 66-byte ASN.1 BITSTRING. Are the values just the first and second half of this bitstring? The private key is an ...
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331 views

How to calculate y value from ((y*y) mod prime) efficiently

i am working ECC-224 bit. can any one tell me, how to calculate y value from ((y*y) mod prime) efficiently for large bit numbers.
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1answer
250 views

Efficient algorithm for remainder calculation over prime field for ECC implementation?

I am working on 224-bit elliptic curve cryptography. In this 224-bit * 224-bit multiplication results 448-bit output. I am reducing 448-bit into prime field range( prime number $2^{224}-2^{96}+1$) ...
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303 views

What crypto system allows for 3 parties: Party 1 who makes an assertion, Party 2 mutates the assertion, Party 3 validates it

I'm looking for the cryptographic equivalent of a Drivers license where the issuer can be verified, the issuer doesn't need to know who you showed the drivers license to, but also allows ...
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Use curve25519 for ElGamal crypto

DJB described curve25519 in his paper which can be found here: http://cr.yp.to/ecdh/curve25519-20060209.pdf. It seems that the main purpose was for Diffie-Hellman key exchange. I think this means that ...
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230 views

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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671 views

Can SRP be used with Elliptic Curves?

I'm sure it can, because SRP (secure remote protocol) can be implemented everywhere where Diffie-Hellman works, but I need a proof to put this aspect into Wikipedia. Edit: ok, can it be at least ...
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261 views

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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1answer
461 views

When do ECC patents end?

As the topic says, since when can ECC cryptography be freely used? Isn't it widely used because of patents? There is no alternative to it on embedded devices and smart cards. Just to mention: I am ...
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1answer
291 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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3answers
361 views

File encryption with one keypair?

I'm working on a program that uses an ECC keypair in a (password protected) PKCS12 file (.pfx) to encrypt files. I like this method because I think it will be higher security (using ECDH to negotiate ...
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129 views

What are unified addition and differential addition in elliptic curve point arithmetic?

A lot of papers use these terms but I do not find a proper explanation of them. Can somebody tell the meaning / difference / intuition / application and if possible with an example.
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855 views

Why do the elliptic curves recommended by NIST use 521 bits rather than 512?

Wikipedia says in reference to the elliptic curves officially recommended by NIST in FIPS 186-3: Five prime fields for certain primes p of sizes 192, 224, 256, 384, and 521 bits. For each of the ...
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1answer
133 views

Discrete log analog of ECM factoring algorithm?

Anecdotally, most factoring algorithms have a corresponding variant algorithm that can be used to attack the discrete log problem using similar ideas. Is there an analog of the elliptic curve (ECM) ...
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1answer
159 views

Generating non-supersingular elliptic curves for symmetric pairings

I am looking into the application of pairings in CPABE in particular. I've notice that the scheme uses a supersingular curve as the basis of the pairing. Looking through Ben Lynn's thesis for the ...
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1answer
135 views

Discrete logs on elliptic curve with embedding degree 3 with the 'MOV' attack

The curve $E(\mathbb{F}_{47}):y^2=x^3+x+38$ has order $61$ and $61|47^3-1$ so the embedding degree of $E$ is $3$ and therefore the MOV attack, presumably using some sort of distortion map and a ...
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307 views

Modulus for elliptic curve point multiplication

I want to implement a point multiplication ($k \cdot P$) operation on FPGA. I have a BN curve $y^2=x^3+2$, and a scalar value $k$. The $x$ and $y$ coordinates of point $P$ are of 256 bits. In the ...
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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 ...
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1answer
224 views

Choosing good parameter for Lenstra's elliptic curve factorization

In Wikipedia, there is an article explaining Lenstra's factorization algorithm. As far as I got it, we choose some $e \in \mathbb{N}$ and a point $P$ on the curve and then calculate $eP$. While ...
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1answer
85 views

tower of extension field

while working on tate pairing, i have to implement towering technique. like i have point p on F(q) and point Q(F(q^k)) (here embedding degree k=12 for BN curve). instead of taking a point Q on ...
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1answer
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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1answer
213 views

Why doesn't this replay attack work on ECDSA?

I've just started working with elliptic curves and ECSDA in particular, so my understanding of the underlying math isn't great. The thing I'm currently stuck on is trying to understand why replay ...
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270 views

ECM Implementation is really slow

I followed the algorithms 14.4 (computes 1st and 3rd coordinates in (X,Y,Z)#k modulo n) and 14.5 (factorization using ECM) in David Bressoud's book 'Factorization and Primality Testing'. I think the ...
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1answer
179 views

Impact of algorithms for factoring using elliptic curves over $\mathbb{Q}$

Recently a few papers have appeared that describe a new approach to factoring, using elliptic curves over $\mathbb{Q}$. See, e.g., Factoring integers and computing elliptic curve rational points, ...
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496 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 ...
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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 ...
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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 ...
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567 views

Why do public keys need to be validated?

For some curves it's necessary to validate the public-key of the other side before running an elliptic-curve Diffie-Hellman key-exchange. Apparently if you don't validate the public key, small ...
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1answer
953 views

BouncyCastle Elliptic Curve implementation

I'm implementing ECDH key exchange in C# using the BouncyCastle library and I'm having a hard time understanding the elliptic curve side (FpCurve). ...
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1answer
321 views

Is there a method to break an EC curve for all key-pairs (Q,d) such that (Q=d*G) faster than breaking every single key-pair?

Related to this question: Is there any memory trade-off that helps such attack? Obviously if the field size is very small (say 40 bits) it´s possible, but what if the field size is 160 bits long? or ...