I've been working with ideal reduction algorithms, and in particular have a need to understand the notion of LLL-reduced along a vector $v$, and in particular what it actually means to have 'small $v$-norm'. All definitions are from Cohen's first computational number theory book (A Course in Computational Algebraic Number Theory, section 6.5.1).
The definition is as follows: Let $K$ be a number field with signature $(r_1, r_2)$. Let $v = (v_i)_{1 \leq i \leq n}$ be a real vector with $v_{r_2 +i} = v_i$ when $ r_1 < i \leq r_1 + r_2$. Define the $v$-norm for $\alpha \in K$ as follows: $$\| \alpha \|_v^2 = \sum_{i=1}^ne^{v_i}| \sigma_i(\alpha)|^2,$$ where the $\sigma_i$ are the field embeddings with their usual ordering. Given an ideal $I$ with $\mathbb{Z}$-basis $\alpha_1, \ldots, \alpha_n$, $$\|\sum_jx_j\alpha_j\|_v^2$$ defines a positive definte quadratic form. Cohen describes in Algorithm 6.5.5 a procedure which takes an ideal basis and vector $v$ as above, computes this quadratic form and LLL reduces the corresponding basis. The algorithm returns $\alpha$, the first element of the LLL reduced basis, and says that it has small $v$-norm.
I am not well versed in the theory of quadratic forms, so I am not sure what the implications of being a short vector in this quadratic form are, or any intuition about what the $v$-norm is trying to capture.
Can we say anything about the distance of the vector $(\sigma_i(\alpha))_i^n$ and the vector $v$?
Any help with understanding, or a pointer to additional resources would be very much appreciated!
First of all, I agree that the concept of the $v$-norm seems quite ad-hoc in the book. Also, the main use case, Algorithm 6.5.6, is completely heuristic. Nevertheless, I hope that I can shed some more light on the intuition.
What is the $v$-norm?
Clearly, the $v$-norm is a scaling of the standard "canonical" $\ell_2$-norm $$ \| \alpha \|_{\mathrm{can}} = \| (\sigma_1(\alpha), ..., \sigma_n(\alpha) \|_2$$ Such a norm is sometimes also called "elliptical", since its norm-1-set is an ellipsoid (while the norm-1-set of the canonical norm is clearly a sphere).
What are the implications of having a vector that is short in the $v$-norm? Mainly, it is Prop. 6.5.4, i.e. a vector $\alpha$ that is a shortest (nonzero) vector for an arbitrary $v$-norm is already a minimum for $I$. What is the point of having a minimum for $I$? I will give my thoughts on that in the next section. In particular, I do not think that there is any relevant relationship between the distance of $(\sigma_i(\alpha))_i$ and $v$.
While it does not seem to be very relevant for Alg. 6.5.6, I also want to mention that these norms appear quite naturally when we scale elements by a constant. In particular, if we let $\gamma \in K \setminus \{0\}$, then $$\| \gamma\alpha \|_{\mathrm{can}} = \sum_i \| \sigma_i(\gamma) \|^2 \| \sigma_i(\alpha) \|^2 = \| \alpha \|_v^2$$ for the vector $v_i = 2\ln\| \sigma_i(\gamma) \|$.
What does Cohen use it for?
As far as I can understand, the main use by Cohen is that by using LLL-reduction based on a random $v$-norm (for a not too unbalanced $v$), we can get a randomized minimal element $\alpha$ of an input ideal $I$. The canonical norm does always give a minimal element as well, but there is just one canonical norm, so it allows us to only find a fixed minimal element for the input ideal $I$.
We care about minimal elements, since for a minimal element $\alpha \in I$, the ideal $I/\alpha$ is "small" in the sense that all $\mathfrak{p}$-adic valuations are small in absolute value compared to the size of $\mathfrak{p}$. In particular, this makes us hope that the $\mathfrak{p}$-adic valuations for large $\mathfrak{p}$ (i.e. those not in the factor base) are zero.
The idea of Algorithm 6.5.6 is now to find sufficiently random ideals $I, J$ such that they are in the same ideal class ($I = \alpha J$) and both $I$ and $J$ are small. In this case, we then hope that both factor over the factor base. As an "optimization", we choose $I$ as a product of factor base elements, hence we only have to hope that $J$ factors. The main point is now, that if we just LLL-reduce $I$ along the "canonical" $v = 0$ norm, we get a suitable $\alpha$ that makes $J = I/\alpha$ small, but it is fixed. In particular, if $J$ does not factor over the factor base, then by using another $v$, we can generate another $J'$ and try again.
A remark on step 6. Note that in step 6 of Algorithm 6.5.6, the $v_i$s are not just the components of the vector $v$, but also used as exponents to generate the ideal $I$ as $$ I = \mathfrak{q} \prod_{1 \leq i \leq s} S_i^{v_i} $$ The vector $v$ determining the norm is then built from $v_i$ where $s < i \leq s + n$. I believe that this is just a way to say that both the exponents of the $S_i$ and the vector $v$ are independently random, but using the letter $v$ for both the exponents and the components of $v$ seems to indicate that there is a relationship that I do not think exists.