# Canonical inclusion map in subfield attack on overstretched NTRU

I'm trying to understand subfield attacks on overstretched NTRU. In the paper https://eprint.iacr.org/2016/127.pdf authors used "canonical inclusion map" to lift vector to full lattice. What does the canonical inclusion map relative to finite fields mean? And where can I read more about this construction?

When $$\mathbb{L}$$ is a subset (or can be view as a subset) of $$\mathbb{K}$$, the canonical inclusion map just means "take an element of $$\mathbb{L}$$ and treat it as element of $$\mathbb{K}$$".
In general, we do it implicitly and it is not necessary to think about it. For example, when we multiply a real number $$r$$ by a complex number $$a + bi$$, we are actually representing $$r$$ as $$r + 0i$$ and using the complex number multiplication.
Most probably, the authors defined this map explicitly to keep things formal and because $$\mathbb{L}$$ and $$\mathbb{K}$$ have different dimensions when view as vector spaces over $$\mathbb{Q}$$, so they want to make sure that there would be no confusion about how to compute the norms and other quantities of elements of $$\mathbb{L}$$ when we use them as elements of $$\mathbb{K}$$.
For exemple, if $$\mathbb{L} := \mathbb{Q}(\sqrt 2)$$ and $$\mathbb{K} = \mathbb{Q}(\sqrt 2, \sqrt 5)$$, then, the elements of $$\mathbb{L}$$ are of the form $$a_0 + a_1\sqrt 2$$ and those of $$\mathbb{Q}$$ are $$a_0 + a_1\sqrt 2 + a_2\sqrt 5 + a_3\sqrt{10}$$. Moreover, $$\mathbb{L}$$ can be seen as a vector space of dimension $$2$$ and $$\mathbb{K}$$ as a vector space of dimension $$4$$ over $$\mathbb{Q}$$.
Thus, it is clear that we can see elements of $$\mathbb{L}$$ as elements of $$\mathbb{K}$$ with $$a_2 = a_3 = 0$$. Thus, when you use the canonical inclusion map from $$\mathbb{L}$$ to $$\mathbb{K}$$, you are also taking vectors of dimension $$2$$ and using them as vectors of dimension $$4$$.
All that said, notice that the authors do not simply use the canonical inclusion map to lift elements of $$\mathcal{O}_{\mathbb{L}}$$ to $$\mathcal{O}_{\mathbb{K}}$$, as they want to guarantee some properties after lifting (like lifted elements being multiples of the secret NTRU vector $$(f, g)$$ and being small).