Table 1 Some notations in this paper

\({\mathbb {F}}_q\)

A finite field of q elements

|

The concatenation operator of vectors or matrices

\(x\xleftarrow {\$}S\)

The operation of selecting an element x from set A uniformly at random

\(||{\mathbf {a}}||\)

The rank of a vector \({\mathbf {a}}\)

\(Len({\mathbf {x}})\)

A function that takes as input a vector \({\mathbf {x}}\), and outputs the length of \({\mathbf {x}}\)

\(Rd_k({\mathbf {x}})\)

A function that takes as input a seed \({\mathbf {x}}\), and outputs a random vector from \({\mathbb {F}}_{q^m}^k\)

\(Conv_t({\mathbf {x}})\)

A function that takes as input a vector \({\mathbf {x}}\) of length \(\lfloor \log _{q^m}\left( {\begin{array}{c}n\\ t\end{array}}\right) \rfloor\) over \({\mathbb {F}}_{q^m}\), and outputs an error vector of length n and weight t. The inverse of function \(Conv_{t}(\cdot )\) is denoted by \(Conv^{-1}_{t}(\cdot )\)

\(Lt_{k}({\mathbf {x}})\)

A function that takes as input a vector \({\mathbf {x}}\), and outputs the left k components

\(Rt_{k}({\mathbf {x}})\)

A function that takes as input a vector \({\mathbf {x}}\), and outputs the right k components