Group-ring Lattice

The Group-ring interpretetion can also be used to construct Lattice for NTRU-like cryptosystems.

Lattice from Group-ring

Let \(G = \lbrace a_1,a_2,\cdots a_n \rbrace\) be a group and \(R\) be a ring. \(R\) is preferably commutative with identity.
The choice of such \(R\) is so that few invertible elements may be present in \(R\).
Let \(RG\) be the group ring. Let \(f,g \in RG\) such that \(f\) is invertible.
Let \(h = f^{-1} * g\) be the Public Key.
Then, \(g = f * h\). We can construct similar matrix equations as we did for NTRU.
\(f = f * 1 + u * 0\) and \(g = f * h + u * q\).
\(H = \begin{bmatrix} h_{a_1^{-1}a_1} & h_{a_1^{-1}a_2} & \cdots & h_{a_1^{-1}a_n} \\ h_{a_2^{-1}a_1} & h_{a_2^{-1}a_2} & \cdots & h_{a_2^{-1}a_n} \\ \cdots \\ h_{a_n^{-1}a_1} & h_{a_2^{-1}a_2} & \cdots & h_{a_n^{-1}a_n} \end{bmatrix}\)
\(\begin{bmatrix} f & g \end{bmatrix} = \begin{bmatrix} f & u \end{bmatrix} * \begin{bmatrix} I_n & H \\ 0_n & qI_n \end{bmatrix}\).

However if the multiplication is from right, i.e., \(f\) and \(g\) are column vectors instead of row vectors,
then the equation and structure of \(H\) changs a little.
\(g = H * f + q * u\) and \(H = [h_{a_ia_j^{-1}}]\) instead of \(H = [h_{a_i^{-1}a_j}]\).

Lattice from twisted Group-ring

Will edit later.