# Questions tagged [elliptic-curves]

Elliptic curves are algebraic-geometric structures with applications in cryptography. Such a curve consists of the set of solutions to a cubic equation over a finite field equipped with a group operation. Questions relating to elliptic curves and derived algorithms should use this tag and might also consider more specific tags such as discrete-logarithm and ecdsa.

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### 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 ...
289 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 ...
646 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 ...
604 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 ...
6k 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 ...
17k views

### How strong is the ECDSA algorithm?

Some cryptographic algorithms are as strong as the size of their key is, while other have some weaknesses that limit their strength (such as SHA-1). How strong is the ECDSA algorithm, and does that ...
342 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 ...
240 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 '$a$' ...
309 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 ...
336 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 ...
301 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 ...
275 views

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

Is there a problem with signing a message like this? ...
1k 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 ...
873 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 ...
254 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 ...
56 views

### 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 ...
2k 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 ...
651 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?
397 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 ?
673 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 ...
178 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?
3k 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 ...
875 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 ...
550 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 ...
957 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 ...
795 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 ...
3k views

### Can one reduce the size of ECDSA-like signatures?

Using $n$-bit ECDSA, a signature has a size of $2·n$. It is possible to recover the public key from this signature, which shows that there is a publicly visible redundancy in the signature. Is ...
239 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) ...
770 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 ...
4k 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, ...
694 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 ...
343 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 ...
247 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 F(q^12)...
596 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 ...
856 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 ...
239 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, ...
2k 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 ...
3k 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 ...
3k 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). ...
833 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 ...
422 views

### Diffie hellman key exchange on elliptic curve over an extension field [closed]

I am attempting to do a final semester project where I implement Diffie-Hellman key exchange on an elliptic curve over an extension field (2^256). Can anybody help me to generate the extension field ...
225 views

### How does DJB's nistp224 manage to fit compressed points into 224 bits?

DJB's nistp224 program purports to be an implementation of elliptic curve Diffie-Hellman relative to the standard NIST P-224 elliptic curve. To the best of my ...
8k views

### Current mathematics theory used in cryptography/coding theory

What are the mainstream techniques borrowed from algebraic geometry (or some other branch of mathematics) which are currently used in cryptography/coding theory? I've only heard about a small subset ...
2k views

### X9.62 Multiplying an elliptic curve point by a number

I'm currently trying to implement ecdsa and the first problem i met -- multiply an elliptic curve point by a number. As far as i understand X9.62 gives some recommendation for doing it but i haven'...
654 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?
2k views

### ECC algorithm pollard's $\rho$ complexity

One of the methods to break a ECDLP is Pollard's rho algorithm. When ECDLP is defined over a finite field $F_p$, and given a relation $S=w.T$, where S and T are a member of $F_p$. Then ECDLP is to ...
2k views

### Standardized parameters for elliptic curve cryptography

When an elliptic curve-based cryptosystem is deployed, a single set of public parameters (consisting of a particular elliptic curve over a finite field as well as a generator of a prime order subgroup ...
4k 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.