## bitcoin – Public key of Elliptic Curve Digital Signature Algorithm

How do I compute my public key, if my private key for ECDSA in SHA-1 is equals to ab2c34b85dd576112f34?

``````where:
x = 54545578718895168534326250603453777594175500187
y = 35454270510029780865563085577751305070431844712
p = 12121157920892373161954235709850086879078532645

``````

## differential equations – How do you find the Inverse of Elliptic İntegral of Second Kind when modulus is large

So I tried to take the inverse of EllipticE when modulus is large, in Mathematica, but the solution gives wrong answer.

``````InverseSeries(Series(EllipticE(x, -k), {x, 0, 12}, {k, Infinity, 1}),y) = InverseFunction(y,k)
``````

For example, I tried `EllipticE(0.5,-9.9) = 0.656` where `x:0.5 , k:-9.9, y:0.656`

But InverseFunction(y,k) is not equal to 0.5. Am I not correctly taking the inverse of the function?
I need a general form of an equation for the inverse of EllipticE. Polynomial approximation is also fine. The approximation should definitely work around when `x-->0` and `k-->-infinity`.
So for the above example, the approximation function result should yield to 0.5 when y=0.656 and k=-9.9. I need to code this function in MCU, so I need an analytical approximation.

Caner

## ag.algebraic geometry – Why is this “the first elliptic curve in nature”?

The LMFDB describes the elliptic curve 11a3 (or 11.a3) as “The first elliptic curve in nature”. It has minimal Weierstraß equation
$$y^2 + y = x^3 – x^2.$$
My guess is that there is some problem in Diophantus’ Arithmetica, or perhaps some other ancient geometry problem, that is equivalent to finding a rational point on this curve. What might it be?

## elliptic curves – why \$ E_1 ( mathbb {Q} _p) approx mathbb {Z} _p \$

I have read an article that says: $$E_1 ( mathbb {Q} _p) approx mathbb {Z} _p$$

Where $$E$$ is an elliptic curve over $$mathbb {Q} _p$$ and $$E_1 ( mathbb {Q} _p) = {P in E ( mathbb {Q} _p): tilde {P} = tilde {O} }$$.

The author says the proof is in "Silverith Arithmetic of Elliptic Curves" on page 191, but it says here:

If $$E$$ is an elliptic curve over $$mathbb {Q} _p$$ and $$has {E}$$ is the formal group, then:

$$E_1 ( mathbb {Q} _p) approx hat {E} (p mathbb {Z} _p)$$

So I don't know a good reference for proving $$E_1 ( mathbb {Q} _p) approx mathbb {Z} _p$$.

## elliptic curves – degree of morphisms and isogenies

$$renewal command { J} { mathrm {Jac}} renewal command { F} { mathbb {F}}$$
I read this paper from B. Gross and there is something I do not understand on p. 22. 945. Here is the context: Fix a prime number $$p equiv 3 pmod 4$$and define the (hyper) elliptic curves via $$F_p$$ given by the (affine) equations
$$X_1: y ^ 2 = x ^ p – x, quad X_2: y ^ 2 = x ^ {p + 1} -1, quad E_1: y ^ 2 = x ^ 3-x, quad E_2: y ^ 2 = x ^ 4-1.$$

I checked (using the Tate isogenicity theorem) that there is non-zero isogenicity $$alpha: J (X_1) to J (X_2)$$ between the jacobian varieties (actually both are isogenic to $$E_1 ^ {(p-1) / 2}$$ about $$F_p$$) and there is non-zero isogenicity $$beta: E_1 to E_2$$.

There is a morphism $$f_2: X_2 to E_2, (x, y) mapsto (x ^ {(p + 1) / 4}, y)$$ that's right now $$(p + 1) / 4$$. Then it is said that this is why we get a morphism $$f_1: X_1 to E_1$$ Degree $$(p + 1) / 4$$, but I don't understand why / how.

I know that $$f_1$$ induces a morphism $$phi_2: J (X_2) to E_2$$We get a morphism $$beta circ phi_2 circ alpha ^ { vee}: J (X_1) to E_1$$hence a morphism $$f_1: X_1 to E_1$$, but I think it has degrees at least the degree of $$f_2$$.
Maybe there is a clever way to compose $$phi_2$$ with other isogenies to maintain the equality of degrees?

Generally with a non-constant morphism $$f_2: X_2 to E_2$$It may not be possible to get a morphism $$f_1: X_1 to E_1$$ to the same extent as $$f_2$$: Just take $$X_2 = E_2 = X_1, f_2 = mathrm {id}$$ and $$E_1$$ an elliptic curve is isogenic but not isomorphic $$E_2$$.
I am probably missing something easy, but I would rather ask for clarification.

## bip 32 hd wallets – Elliptic Curve Point at Infinity

What exactly is the "point in infinity"?

It is a point added to the points on the curve. Together they form a group. It has the following characteristics:

• (x, y) + (x, -y) = infinite
• (x, y) + infinity = (x, y)
• infinite + (x, y) = (x y)

Is there more than one "point in infinity"?

Only one.

How can I tell if my x and y generated by EC are the "point of infinity"?

The point at infinity is not on the curve, so it has no x or y coordinates. It is displayed whenever a point is added to its own negation (to which the normal addition rule has no answer).

Is there a way to calculate the "point of infinity"?

It is simply "the point in infinity", there is nothing to calculate.

## ag.algebraic geometry – the extended upper half-plane \$ mathcal {H} ^ * \$ as stable elliptic curves

Write $$mathcal {M} (1)$$ (and its compactification $$overline { mathcal {M}} (1)$$) for the module space (stable) elliptic curves. Display both as a stack $$text {To} _ text {et}$$, analytical spaces with the etale topology.

Consider the universal family of elliptic curves and stable elliptic curves and give a pullback diagram
$$requires {AMScd}$$
$$begin {CD} mathcal {H} @ >>> overline { mathcal {H}} \ @V V V @VV V \ mathcal {M} (1) @ >>> overline { mathcal {M}} (1) end {CD}$$
what exhibits $$mathcal {M} (1) simeq ( mathcal {H} / text {SL} _2 ( mathbf {Z}))$$ as a global quotient of analytical space $$mathcal {H}$$. They are also called quotient stacks $$overline { mathcal {H}} / Gamma (N) = X (N)$$ to the $$N> 2$$.

On the other side is the extended upper half level $$mathcal {H} ^ * = mathcal {H} cup mathbf {P} ^ 1 ( mathbf {Q})$$, that is a Hausdorff topological space on which $$text {SL} _2 ( mathbf {Z})$$ looks really discontinuous and $$mathcal {H} ^ * / Gamma (N) simeq X (N)$$ as topological spaces.

question: is $$overline { mathcal {H}}$$ an analytical space? What is their relationship with $$mathcal {H} ^ *$$ (what i think is Not an analytical space, only its quotients are according to congruence subgroups)?

## arithmetic geometry – elliptic curves and their neron model

To let $$E$$ be an elliptic curve over $$mathbb {Q}$$, For a prime number $$p$$, To let $$mathcal {E} _p$$ designate his neron model over $$mathbb {Z} _p$$, Let also $$Phi_p (E)$$ denote the component group of $$mathcal {E} _p$$,

The structure of $$Phi_p (E)$$ is known and I want to study it if $$E$$ added multiplicative reduction $$p$$, First, if $$E$$ split multiplicative reduction at $$p$$, then it is known that $$Phi_p (E) simeq mathbb {Z} / {n mathbb {Z}}$$, Where $$n$$ is the (normalized) $$p$$-adic assessment of the discriminant of $$E$$, (I admit this fact.) Next, when $$E$$ has an undivided multiplicative reduction $$p$$, then $$Phi_p (E) simeq mathbb {Z} / {m mathbb {Z}}$$, Where $$m = 1$$ or $$2$$ so that $$m equiv n pmod 2$$,

Here's my question: Suppose that $$E$$ has an undivided multiplicative reduction $$p$$, As above, we use the same notation ($$m$$. $$n$$. $$Phi_p (E)$$ Etc).
We know the following.

1. There is an unbranched quadratic extension $$L / { mathbb {Q} _p}$$ so that $$E / { mathcal {O} _L}$$ divided multiplicative reduction.

2. The Neron model does not change when the base changes.

If the two above are correct, then the component group $$Phi_p (E)$$ can be calculated using the neron model of $$E / L$$, Where $$E$$ divided multiplicative reduction. Since the $$p$$-adic rating does not change under the unchanged extension, the component group of the Neron model of $$E / L$$ is isomorphic to $$mathbb {Z} / {n mathbb {Z}}$$, Consequently, $$Phi_p (E)$$ is isomorphic too $$mathbb {Z} / {n mathbb {Z}}$$what is wrong if $$n> 2$$,

I don't know where my argument fails. Please correct me!

## Geometry – problem with equating the coefficient of the elliptic curve

To let, $$E: = y ^ 2 = x ^ 3 + Ax + B$$ an elliptic curve, 2 points $$P, -Q$$ on $$E$$ so that $$2P = -Q$$, we can write

$$y ^ 2-x ^ 3 + Ax + B = (x – e_1) (x – e_2) ^ 2 = 0$$
Where, $$e_1 = x (Q), e_2 = x (P)$$ a double root. So,
$$y = m (x-x (Q)) + y (Q)$$, Here $$y$$ is the line that goes thoroughly $$P, Q$$, and $$m$$ is the slope.

$$(x – e_1) (x ^ 2 – 2xe_2 + e_2 ^ 2)$$
$$= (x ^ 3 – 2x ^ 2e_2 + xe_2 ^ 2) – (x ^ 2e_1 – 2xe_1e_2 + e_1e_2 ^ 2)$$
$$= x ^ 3 – 2x ^ 2e_2 + xe_2 ^ 2-x ^ 2e_1 + 2xe_1e_2-e_1e_2 ^ 2$$
$$= x ^ 3 + x ^ 2 (-1) (2e_2 + e_1) + x (e_2 ^ 2 + 2e_1e_2) + (- 1) e_1e_2 ^ 2$$

Of $$y ^ 2-x ^ 3 + Ax + B = 0$$ we get coefficients of $$x ^ 2$$ is $$(- 1) m ^ 2$$, We find the equality coefficient, $$m ^ 2 = (2e_2 + e_1)$$,

The problem is, if I simplify both sides and then equate the coefficient, I get the correct coefficient of $$x ^ 2$$ but not for $$x$$ (if plug-in value of $$P, Q$$, the coefficients of $$x ^ 2$$ just doesn't satisfy $$x$$).

What's the problem?

## Cryptography – private key for generating public keys bitcoin vb.net reports unknown error – secipt256k1 elliptic curve

I try to use `secp256k1 eliptic curve` to get that `public key` of `private key` but it gives me one `unknown error` error

`````` Private Sub Button13_Click(sender As Object, e As EventArgs) Handles Button13.Click
Dim private_key = "68040878110175628235481263019639686"
'public key should be Nr6MbFUfMovKCX4vd5YpQnRYsN4rq6pNPEEBKmicAEwwuYLpJrt5LsRvfvR2G8pJ5rMchEMWDYJ7rdYGY7PjxHEa
Dim public_key As String

Using eliptic As New ECDsaCng()
eliptic.HashAlgorithm = CngAlgorithm.ECDsaP256

Dim data() As Byte = Encoding.UTF8.GetBytes(private_key)
Dim key As Byte() = eliptic.SignData(data)
public_key = key.ToString
End Using

TextBox15.Text = public_key
End Sub
``````