user129654: From how I understand it, the equation for the public key is defined as so:

K = k * G

With K being the public key, k the private key and G the generator point.

Is G a constant? (as far as I read, it is a constant.)
If it is, how is it not possible to simply do K/G = k?
If it is not, how is it created out of the private key?


user30267: Mathematicians like their analogies, sometimes those analogies can be helpful, but they can equally be confusing, particularly to those new to a field.
In mathematics a "group" consists of a set of elements and an operator that satisfies some requirements. The operator must be associative, there must exist an identity element, and for every element there must exist an inverse element under the operator.
Some groups are written by analogy to addition, that is the group operator is represented by a "+" sign, and the identity element is represented by "0". Other groups are written by analogy to multiplication, that is the group operator is written as a multiplication operation (e.g. "*","." or nothing at all) and the identity element is represented by "1". Elliptic curves are traditionally written by analogy to addition.
We can conceive of applying the group operator repeatedly to k copies of G. If the group is written by analogy to addition then this is analogous to multiplication, if the group is written by analogy to multiplication then this is analogous to exponentiation.
So k * G refers not to regular multiplication but to an operation that is in a particular way analogous to multiplication.
You might naively think that this would take time proportional to k, but thanks to the associativity we can calculate it in time proportional to the logarithm of k. So we can quickly calculate k * G even for very large k.
Reversing the operation is not so easy though. The problem is known as the "Elliptic curve discrete logarithm problem" and the best known methods have a complexity proportional to roughly the square root of the number of elements in the group.
Why is it called the "Elliptic curve discrete logarithm problem" when it's more analogous to division than to a logarithm in the notation typically used for elliptic curves? because it's a variant of a similar problem studied for a group that is traditionally written by analogy to multiplication.

user208: 
How I understood it, the equation for the public key is defined as so K = k * G
With K being the public key, k the private key and G the generator point.

That's correct.

Now my first question, is G a constant? (as far as I read, it is a constant)

It is. It is the point with coordinates (55066263022277343669578718895168534326250603453777594175500187360389116729240,  32670510020758816978083085130507043184471273380659243275938904335757337482424). You can verify that these coordinates satisfy the curve equation: Y^2 = X^3 + 7 (mod 115792089237316195423570985008687907853269984665640564039457584007908834671663).

And if it is, how is it not possible to simply do K/G = k?

The answer is simply that we don't know of an efficient way to do "division" for elliptic curves (which is called the discrete logarithm problem), and it is assumed no such way can be found. The best known algorithm for doing this for this specific curve takes around 2128 iterations, an impossibly large amount.
You have to be aware that the * in k * G as you write it is no ordinary multiplication (what would that even mean for points?). It is a rather complicated operation, and people have figured out useful properties of it which are somewhat similar to multiplication (it forms a cyclic group, so the same symbol is used). But nobody ever found a way to perform the reverse operation, and this became the basis for elliptic curve cryptography.

129654: <p>From how I understand it, the equation for the public key is defined as so:</p>
<blockquote>
<p><code>K = k * G</code></p>
</blockquote>
<p>With K being the public key, k the private key and G the generator point.</p>
<ul>
<li>Is G a constant? (as far as I read, it is a constant.)</li>
<li>If it is, how is it not possible to simply do K/G = k?</li>
<li>If it is not, how is it created out of the private key?</li>
</ul>
 30267: <p>Mathematicians like their analogies, sometimes those analogies can be helpful, but they can equally be confusing, particularly to those new to a field.</p>
<p>In mathematics a &quot;group&quot; consists of a set of elements and an operator that satisfies some requirements. The operator must be associative, there must exist an identity element, and for every element there must exist an inverse element under the operator.</p>
<p>Some groups are written by analogy to addition, that is the group operator is represented by a &quot;+&quot; sign, and the identity element is represented by &quot;0&quot;. Other groups are written by analogy to multiplication, that is the group operator is written as a multiplication operation (e.g. &quot;*&quot;,&quot;.&quot; or nothing at all) and the identity element is represented by &quot;1&quot;. Elliptic curves are traditionally written by analogy to addition.</p>
<p>We can conceive of applying the group operator repeatedly to k copies of G. If the group is written by analogy to addition then this is analogous to multiplication, if the group is written by analogy to multiplication then this is analogous to exponentiation.</p>
<p>So <code>k * G</code> refers not to regular multiplication but to an operation that is in a particular way analogous to multiplication.</p>
<p>You might naively think that this would take time proportional to k, but thanks to the associativity we can calculate it in time proportional to the logarithm of k. So we can quickly calculate <code>k * G</code> even for very large k.</p>
<p>Reversing the operation is not so easy though. The problem is known as the &quot;Elliptic curve discrete logarithm problem&quot; and the best known methods have a complexity proportional to roughly the square root of the number of elements in the group.</p>
<p>Why is it called the &quot;Elliptic curve discrete logarithm problem&quot; when it's more analogous to division than to a logarithm in the notation typically used for elliptic curves? because it's a variant of a similar problem studied for a group that is traditionally written by analogy to multiplication.</p>
 208: <blockquote>
<p>How I understood it, the equation for the public key is defined as so <code>K = k * G</code>
With K being the public key, k the private key and G the generator point.</p>
</blockquote>
<p>That's correct.</p>
<blockquote>
<p>Now my first question, is G a constant? (as far as I read, it is a constant)</p>
</blockquote>
<p>It is. It is the point with coordinates <code>(55066263022277343669578718895168534326250603453777594175500187360389116729240,  32670510020758816978083085130507043184471273380659243275938904335757337482424)</code>. You can verify that these coordinates satisfy the <a href="https://en.wikipedia.org/wiki/Elliptic_curve" rel="noreferrer">curve</a> equation: <code>Y^2 = X^3 + 7 (mod 115792089237316195423570985008687907853269984665640564039457584007908834671663)</code>.</p>
<blockquote>
<p>And if it is, how is it not possible to simply do K/G = k?</p>
</blockquote>
<p>The answer is simply that we don't know of an efficient way to do &quot;division&quot; for elliptic curves (which is called the <a href="https://en.wikipedia.org/wiki/Discrete_logarithm_records#Elliptic_curves" rel="noreferrer">discrete logarithm</a> problem), and it is assumed no such way can be found. The best known <a href="https://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm_for_logarithms" rel="noreferrer">algorithm</a> for doing this for this specific curve takes around 2<sup>128</sup> iterations, an impossibly large amount.</p>
<p>You have to be aware that the <code>*</code> in <code>k * G</code> as you write it is no ordinary multiplication (what would that even mean for points?). It is a rather <a href="https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication" rel="noreferrer">complicated operation</a>, and people have figured out useful properties of it which are somewhat similar to multiplication (it forms a <a href="https://en.wikipedia.org/wiki/Cyclic_group" rel="noreferrer">cyclic group</a>, so the same symbol is used). But nobody ever found a way to perform the reverse operation, and this became the basis for <a href="https://en.wikipedia.org/wiki/Elliptic-curve_cryptography" rel="noreferrer">elliptic curve cryptography</a>.</p>
