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…on; fixes with sections indexing
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| Notice that while the basis in not unique for the given vector space, the number of basis vectors was the same. | ||
| This raises the natural question: can we find the basis $\mathbf{e}_1,\dots,\mathbf{e}_m$ for $\mathbb{F}^n$ or $\mathbb{F}[x]^{<n}$ with $m \neq n$? | ||
| This raises the natural question: can we find the basis $\mathbf{e}_1,\dots,\mathbf{e}_m$ for $\mathbb{F}^n$ or $\mathbb{F}[x]^{\leq n-1}$ with $m \neq n$? |
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I would leave
| Consider the group of integers $(\mathbb{Z},+)$ and the group of integers modulo $12$ $(\mathbb{Z}_{12},+)$. The function $\pi: \mathbb{Z} \to \mathbb{Z}_{12}$ defined as $\pi(x) \equiv x \;\text{mod}\; 12$ is a homomorphism. Indeed: | ||
| \begin{xequation*} | ||
| \pi(a+b) = (a+b) \;\text{mod}\; 12 = (a \;\text{mod}\; 12) + (b \;\text{mod}\; 12) = \pi(a) + \pi(b). | ||
| \pi(a+b) \equiv (a+b) \;\text{mod}\; 12 \equiv (a \;\text{mod}\; 12) + (b \;\text{mod}\; 12) \equiv \pi(a) + \pi(b). |
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Meh, I still feel weird about this example.
I mean, in general algebra, once you have a ring
So writing
Then,
Moreover, one can show
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I think \pi(a) + \pi(b) is a bit confusing in general, because the + on the
right-hand side is addition in Z_12, which already includes the same mod-12
reduction that \pi itself performs. The reader is left wondering whether the
line is correct or whether there is a catch. I reverted to the previous version in 900e5d6.
ZamDimon
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ZamDimon
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Good, thank you @DarkWindman!
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| Now, the question is: how do we implement the scalar multiplication $[k]P$ for the given scalar $k \in \mathbb{Z}_r$ and point $P \in E(\overline{\mathbb{F}}_q)$? | ||
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| First idea: let us simply add $P$ to itself $k$ times. Well, the complexity would be $O(k)$ in this case, which is even harder than solving the discrete logarithm problem (recall that the discrete logarithm problem has a complexity of $O(\sqrt{k})$). Yikes. |
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Objective
This PR is primarily intended for finalizing the current state of the book: fixing weird formulations in existing chapters, fixing typos and figures' misalignments etc.
Changed
hyperrefsettings: previous were cringy