# Tagged Questions

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 ...

52 views

### Order and cofactor of the base point? [duplicate]

What is the order and cofactor of a base point? Is it possible to deduct the order and cofactor, given just the basepoint. What about the other way around from order and cofactor to basepoint?
193 views

### Choosing an optimal generator for an irreducible polynomial over a binary field?

I am reading the Certicom tutorial βAn Example of an Elliptic Curve Group over F2m β and I have following questions: How do they assume that generator $g = (0010)$ is correct for this polynomial? ...
38 views

### Key sizes for RSA over elliptic curves

I know that it is possible to define RSA over elliptic curves just as DSA and Diffie-Hellman have been. I know that it doesn't offer much of a speed advantage, but does it at least reduce the size of ...
80 views

### Construction of division polynomials

I'm trying to understand the construction of the division polynomials used in Schoof's algorithm. I firstly followed this report of Charlap and Robbins. I stuck with the definition of the leading ...
41 views

### Is there any place where I can find test vectors for point addition and doubling of ECC?

I want to extensively test my implementation of point addition and doubling. I have only one test vector with me. I need more values to test. In the web, I could find test vectors only for key pair ...
71 views

### Scalar multiplication with projective coordinates

I'm implementing point addition, doubling and scalar multiplication using projective coordinates. I took reference from this link https://www.nsa.gov/ia/_files/nist-routines.pdf I have implemented ...
57 views

### Simulation of a custom build network security algorithm with ElGamal Cryptosystem using Elliptic Curve

I am trying to build an algorithm to encrypt and decrypt text using ElGamal Cryptosystem using Elliptic Curve. My algorithm generation is done. But at simulation part I stuck. My algorithm steps are ...
387 views

### ECDSA signature verifiable 1-way transformations

Alice signs a message $m$ with her private key, yielding a signature ($r$,$s$). I want to prove to someone else that I have this signature, but I don't want them to have the knowledge of what ($r$,$s$...
26 views

### Do most TLS 1.2 implementations express curves in a canonical form when performing EC arithmetic?

Sorry if this is a silly question, but does anyone know if the cryptographic libraries which implement TLS 1.2 for Firefox, Chrome, etc. express a given curve in a canonical form (i.e. one of ...
35 views

### Can the backdoor in Dual_EC_DRBG be used to create a public key stream cipher?

Dual_EC_DRBG has the property that if $Q = e\cdot P$, someone who knows $e$ can break the PRNG. This seems to lead to a public-key stream cipher: Alice chooses a random $P, e$, where $P$ is a ...
101 views

### SHA1 collisions and the impact for ECDSA signatures

What will it mean for ECDSA using SHA1 when we have practical attacks breaking the collision resistance property of SHA1? [UPDATE] Added a bit more details to be clear. If $(r,s)$ is the ECDSA ...
7k 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 ...
453 views

### How to determine the order of an elliptic curve group from its parameters?

Let $\quad E:\; y^2 = x^3 + ax + b \quad$ be an elliptic curve defined over a finite field $\mathbb F_q$ where $q = p^n$, $a,b \in \mathbb F_q$ and $p \neq 2, 3$. By Hasse's theorem we know that the ...
93 views

### Degrade in performance with SSL_OP_SINGLE_ECDH_USE?

We have used SSL_OP_SINGLE_ECDH_USE when setting up our SSL_CTX . This seems to be causing a degrade in performance. I'm not able to find proper documentation for it except that it generates new ...
207 views

### Difference between Pseudo Mersenne primes and Generalized Mersenne primes

The field prime numbers $p$ proposed by the NIST standards are referred to as Generalized Mersenne prime numbers [1] and as Pseudo Mersenne prime numbers [2]. Is there a difference between Pseudo ...
2k views

### ECDSA Compressed public key point back to uncompressed public key point

From the ECDH demo here, if I generate a private key for Alice I can get _ P = 1175846487558108474218546536054752289210804601041 Which gives the following public ...
46 views

### Example of Projective Coordinates

Given the affine form of coordinates $(x,y)$ such as $(5,3)$, if I want to convert $(5,3)$ to projective coordinates $(x,y,z)$, should the form of point be $(5,3,1)$? It is triplet not a point, right? ...
113 views

### Convert projective to affine coordinates in ECC? [closed]

I am working with my project. I use projective coordinates but when I convert to affine coordinates, I can't get it. Can anyone help me? Projective Coordinates $(X,Y,Z)$ to Affine Coordinates $(X,Y)$:...
119 views

### Elligator-2 against curves over Fq, q mod 4 = 3

It appears that the conditions for applicability of Elligator-2 against many of the SaveCurves curves, where $q \mod 4 = 3$ will inevitably poke a hole in the bit-string set over $(0, 1, .. (q-1)/2)$. ...
80 views

### Semaev summation polynomials

I am little confused how this attack works. We have the points $P, Q$ such that $Q = nP$. We let $u_{1}$and $u_{2}$ such that $R(x,y)=u_{1}P+u_{2}Q$. Then if we find the solution $x_1,...,x_n$ of the ...
152 views

### Limitations of Elliptic Curve Cryptography?

Simple question, what are the limitations of ECC, both in terms of application and how secure it is? I heard that the NSA were able to read emails a few years back due to a backdoor they had ...
34 views

### Using Montgomery ladder to calculate the coordintes

In one of my assignments I need to solve the below: For a Montgomery curve $3v^2 = u^3+u^2+u$ over ${\mathbb{F}}_{11}$ and point $P = (9,8)$. I need to compute $x$ coordinate of $3P$ using ...
64 views

### Optimal same-base exponentiation?

I've (finally) implemented the answer to this question in our library, which stated how to transform montgomery curves (and points) to weierstrass curves (and points). Now, for scalar multiplication, ...
81 views

### Can we break ECDLP with this machine?

Let $P$ and $Q$ are two points of NIST elliptic curve $E$ (defined over $F_{2^m}$ with prime $m$) and $k$ is a private key such that $k.P=Q$. Also we have a machine that is able to leak some ...
97 views

### Performance of ECDSA, ECKCDSA and ECGDSA

It is proven that ECDSA algorithms are faster in key and signature generation compared to RSA. In addition, the signatures are much shorter. However, I would like to know the performance difference ...
41 views

### Elliptic ElGamal Public Key Cryptosystem doubt

I need an example of Elliptic ElGamal Public Key Cryptosystem. I have been trying with some values but I don't get the right solution. I have $p=13$, the elliptic curve $E:y^2=x^3+11x+7$ and a point ...
61 views

### Convert messages to elliptic curve points [duplicate]

Let $E$ be an elliptic curve; $\alpha,\beta$ two points of $E$; and $a$ a private key such that $\beta=a\cdot\alpha$. We choose random integer $k$ and plain text $x\in E$. Encryption and decryption ...
123 views

### Counting points on elliptic curve over binary field

How to count number of rational points on elliptic curve over binary field?
253 views

### Why is 2048-bit RSA always paired with 320-bit ECC?

You may already have noticed that most smart cards ship with 2048-bit RSA support and 320-bit ECC over GF(p) support. You may have already asked yourself "why exactly 320-bit?". Now I remember having ...
118 views

### Is signing a plaintext sufficient?

If I have N bytes of plaintext, does signing it with my private key prove (to holders of my public key) that I have signed that exact plaintext messages? i.e. could an attacker use the plaintext and ...
55 views

271 views

### How do I get the equivalent strength of an ECC key?

I know how to calculate the comparable symmetric strength of an RSA modulus: calculate the running time for a field sieve. This is how NIST gives approximate symmetric sizes for asymmetric algos in ...
433 views

### Koblitz encoding a message to a point, what is the “associated auxiliary base parameter”?

I am looking at the Koblitz method for encoding a message as an elliptic curve point. The first step given in the paper I'm reading is: "Choose an elliptic curve and its associated auxiliary base ...
99 views

### HD (Hierarchical Deterministic) Keys using Safe Curves?

Bitcoin's HD (Hierarchical Deterministic) Keys as described in BIP32 allow for a master key to be created (a private key and a chain code) such that a tree of both public and private keys can be ...