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# Basic Introduction¶

## Lattice Definition¶

The lattice is the linear combination of all integer coefficients of n ($m\geq n$) linearly independent vectors $b_i(1\leq i \leq n)$ of the m-dimensional Euclidean space $R^m$, ie $L(B)=\{\sum\limits_{i=1}^{n}x_ib_i:x_i \in Z,1\leq i \leq n\}$

Here B is a collection of n vectors, we call

• These $n$ vectors a set of bases of the lattice $L$.
• The rank of the lattice $L$ is $n$.
• The number of bits in $L$ is $m$.

If $m = n$, then we call this format full rank.

Of course, the space can be other groups instead of $R^m$.

## Basic Definition in Lattices¶

### Successive Minimum¶

Let lattice $L$ be a lattice in the m-dimensional Euclidean space $R^m$ with rank $n$, then the continuous minimum length of $L$ (successive minima) is $\lambda_1,...,\lambda_n \in R$, where for any $1 \leq i\leq n$, $\lambda_i$ is the minimum value to satisfy that for $i$ linearly independent vectors $v_i$, $||v_j||\leq \lambda_i,1\leq j\leq i$.

Obviously we have $\lambda_i \leq \lambda_j ,\forall i

## Calculating Difficult Problems in the Lattice¶

Shortest Vector Problem (SVP): Given the lattice L and its base vector B, find the non-zero vector v in the lattice L such that for any other non-zero vector u in the lattice, $||v| | \leq ||u||$.

$\gamma$-Approximate Shortest Vector Problem (SVP-$\gamma$): Given a fixed L, find the non-zero vector v in the lattice L such that for any other non-zero vector u in the lattice, $|| v|| \leq \gamma||u||$.

Successive Minima Problem (SMP): Given a lattice L of rank n, find n linearly independent vectors $s_i$ in lattice L, satisfying $\lambda_i(L)=||s_i| |, 1\leq i \leq n$.

Shortest Independent Vector Problem (SIVP): Given a lattice L of rank n, find n linear independent vectors $s_i$ in lattice L, satisfying $||s_i|| \leq \lambda_n(L), 1\leq i \leq n$.

Unique Shortest Vector Problem (uSVP-$\gamma$): Given a fixed L, satisfying $\lambda_2(L) > \gamma \lambda_1(L)$, find the shortest vector of the cell.

Closest Vector Problem (CVP): Given the lattice L and the target vector $t\in R^m$, find a non-zero vector v in a lattice such that for any non-zero vector u in the lattice , satisfy $||vt|| \leq ||ut||$.