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7

It happens that a line usually (not always) cuts three points in a elliptic curve by the Bezout theorem. This is the case for the points and the curve you are asking for. So the sum of two points are defined like the inverse of the third point intercepted by the line that cut $P$ and $Q$ (let's name it $R$). So we need to find $-R$ because $P+Q=-R$ by ...


6

The tangent line at point $P(x_1,y_1)$ is: $\frac{\partial f}{\partial x}(x-x_1)+\frac{\partial f}{\partial y}(y-y_1)$ Solving differentials, we obtain: $\frac{\partial f}{\partial x}=3x^2+2x$ $\frac{\partial f}{\partial y}=-2y-1$ For $x_1=0$ and $y_1=0$ we have: $\frac{\partial f}{\partial x}=3x^2+2x = 0$ $\frac{\partial f}{\partial y}=-2y-1 = -1$ ...


4

It mainly depends on how the algorithm was selected. If it was selected by a public competition like for AES, then it is likely to be secure. If it was forced in by the NSA such as Dual-EC random number generator, then you may have some doubts. Other questions you may want to ask yourself are: Is this an "original" algorithm or was the problem that it ...


3

Feasible? Sure, there are lattice algorithms that are competitive in performance with RSA. However, there are drawbacks, like: They've been studied less than RSA or ECC, especially the individual algorithms. The most well studied system, NTRU, is patented. No generic proof that I know of that there isn't a quantum algorithm to solve them. The first one ...


3

If you want $N$ serial numbers, your serial numbers will have to use $n$ bits for uniqueness, where $n = \log_2 N$. So if you have 100 bits to use for the serial, you could use 20 to get about a million serials and have 80 bits to use for a cryptographic MAC or signature. Now there are two approaches, the symmetric and the asymmetric. In the symmetric ...


2

The cyclic group over the ECDLP problem is posed is a subset of the set of point of the elliptic curve. That is to say, not all the points in the referred curve will be in the cyclic group. What you've called as $G$ confuses with another notation of the generator of a cyclic group $<G>=\{G,[2]G,\cdots,[n]G=\mathcal{O}\}$. This is the meaning of what ...


2

Trevor Perrin wrote a library doing exactly that. Explanation can be found on in the curves mailing list archives. To convert a Curve25519 public key $x_C$ into an Ed25519 public key $y_E$, with a Ed25519 sign bit of $0$: $$y_E = \frac{x_C - 1}{x_C + 1} \mod 2^{255}-19$$ The Ed25519 private key may need to be adjusted to match the sign bit of $0$: if ...


2

Your second equation seems a bit off. In the curve of equation: $$ y^2 + cy = x^3 + ax + b $$ in a binary field $\mathbb{F}_{2^m}$, to add point $P_1 = (x_1,y_1)$ to point $P_2 = (x_2,y_2)$, resulting in point $P_3 = (x_3,y_3)$, then the two equations are: \begin{eqnarray*} x_3 &=& \lambda^2 + x_1 + x_2\\ y_3 &=& \lambda (x_1+x_3) + y_1 + c ...


1

Related to Curve25519 Curve25519 seems to be secure so far. Yet, you have to remind yourself that Dr. Bernstein specified Curve25519 for key-exchange. Meaning: key-generation, transaction signing, and verification are somewhat different beasts – you might want to cross-check on that before jumping toward Curve25519. Sure, Curve25519-java supports signing… ...


1

There are good reasons to think an algorithm being in Suite B is evidence NSA thinks it's secure (they are used to protect classified materials). There are also reasons to think algorithms they recommend for others may not be (it's happened before). So I don't think you can objectively say much about an algorithm either way just on the basis of whether it's ...


1

Support for this has now (2014-08-05) been added to Sodium. Implement ed25519 -> curve25519 keys conversion Look at test/default/ed25519_convert.c for some example code.


1

I have figured out a way to use TLS with only a EC key by using DSA instead of RSA. I had not realized you could do DSA with a EC key. My mistake was trying to use RSA to sign the certificate. Now I can generate my certificate and self-sign it only with the EC key. The peer will simply check to make sure the certificate was signed by the appropriate node ID ...


1

A classical resut is that over a finite field $\mathbb{F}_p$ the group $E(\mathbb{F}_p)$ is either cyclic or isomorphic to the product of two cyclic groups. For the order of the group see this question. For cryptographic purposes, i.e., when you require that the ECDLP is hard, you will firstly have to rule out some weak curves (supersingular or anomalous ...



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