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## 数学代写|数学建模代写math modelling代考|Extension Field Type

The central map of MI (Sect. 2.1, [69]) is constructed by a univariate monomial over an extension field. While MI was already broken, the idea generating $G$ over an extension field is used for several MPKCs. The central map $G: k^{n} \rightarrow k^{m}$ of such an MPKC is generally described as follows.

Let $r \geq 1$ be a common divisor of $n$ and $m, N:=n / r, M:=m / r, K$ an $r$ extension of $k$ and $\left{\theta_{1}, \ldots, \theta_{r}\right} \subset K$ is a basis of $K$ over $k .$ Denote by $\phi_{N}: k^{n} \rightarrow$ $K^{N}$ is a one-to-one map, e.g. $\phi_{N}\left(x_{1}, \ldots, x_{n}\right)=\left(x_{1} \theta_{1}+\cdots+x_{r} \theta_{r}, \ldots, x_{n-r+1} \theta_{1}+\right.$ $\cdots+x_{n} \theta_{r}$ ) for $x_{1}, \ldots, x_{n} \in k$, and define a polynomial map $\mathscr{B}: K^{N} \rightarrow K^{M}$ to be inverted feasibly. The central map $G$ is constructed by $G:=\phi_{M}^{-1} \circ \mathscr{G} \circ \phi_{N}$.
$$G: k^{n} \stackrel{\phi_{N}}{\rightarrow} K^{N} \stackrel{\mathscr{G}}{\rightarrow} K^{M} \stackrel{\phi_{M}^{-1}}{\longrightarrow} k^{m} .$$
It is known that the polynomials $g_{1}(x), \ldots, g_{m}(x)$ in $G(x)$ are quadratic forms of $x=\left(x_{1}, \ldots, x_{n}\right)^{t} \in k^{n}$ over $k$ if and only if the polynomials $\mathscr{G}{1}(X), \ldots, \mathscr{G}{M}(X)$ in $\mathscr{G}(X)$ are quadratic forms of $\bar{X}:=\left(X_{1}, \ldots, X_{N}, X_{1}^{q}, \ldots, \ldots, X_{N}^{q^{r-1}}\right)^{t}$ over $K$. It is because the one-to-one map $\phi_{N}$ is given by the matrix $\Theta_{N}:=\left(\theta_{j}^{q^{i-1}} \cdot I_{N}\right){1{N}$ is the identity matrix of size $N$. In fact, if $X=\left(X_{1}, \ldots, X_{N}\right)^{t}:=\left(x_{1} \theta_{1}+\right.$ $\left.\cdots+x_{r} \theta_{r}, \ldots, x_{n-r+1} \theta_{1}+\cdots+x_{n} \theta_{r}\right)^{t}$, it holds
$$\Theta_{N} x=\bar{X}$$
Then $F$ and $G$ have the relation \begin{aligned} F(x)=&\left(T \circ \Theta_{M}^{-1}\right) \cdot\left(\mathscr{G}{1}\left(\phi{N}(S(x))\right), \ldots, \mathscr{G}{N}\left(\phi{N}(S(x))\right)\right.\ &\left.\mathscr{G}{1}\left(\phi{N}(S(x))\right)^{q}, \ldots, \mathscr{G}{N}\left(\phi{N}(S(x))\right)^{q^{r-1}}\right)^{t} \end{aligned}
and $\mathscr{G}{i}\left(\phi{N}(S(x))\right)^{q^{j}}$ is written by
$$\mathscr{G}{i}\left(\phi{N}(S(x))\right)^{q^{j}}=X^{t}\left(\Theta_{N} S \Theta_{N}^{-1}\right)^{t} G_{i}^{\left(q^{j}\right)}\left(\Theta_{N} S \Theta_{N}^{-1}\right) X+(\text { linear form of } X)$$
for some $n \times n$ matrix $G_{i}^{\left(q^{\prime}\right)}$ with $K$-entries. The matrix $G_{i}^{\left(q^{\prime}\right)}$ is important for the security of the extension field type MPKCs.
In this subsection, we describe several examples of such MPKCs.

## 数学代写|数学建模代写math modelling代考|Hidden Field Equation

Hidden Field Equation $(H F E)$ proposed by Patarin [79] is constructed with $n=m=$ $r$ (namely $N=M=1$ ) and
$$\mathscr{G}(X)=\sum_{0 \leq i \leq j \leq d} \alpha_{i j} X^{q^{i}+q^{j}}+\sum_{0 \leq i \leq d} \beta_{i} X^{q^{i}}+\gamma,$$
where $1 \leq d \ll n$ is an integer and $\alpha_{i j}, \beta_{i}, \gamma \in K$. The decryption of HFE is obtained by solving a univariate polynomial equation $\mathscr{G}(X)-Y=0$ of degree $D \leq 2 q^{d}$. Its complexity is $O\left(D^{3}+n D^{2} \log q\right)$ by the Berlekamp algorithm [8, 9].

For the security of HFE, it has been reported that $F$ of HFE with small $d$ is inverted efficiently by the Gröbner basis attack [45]. It is known that the degree of regularity of the corresponding polynomial system is bounded by $\frac{1}{2}(q-1)\left\lfloor\log {q}\left(2 q^{d}-1\right)+\right.$ $1\rfloor+2$ [34, 50]. Furthermore, since the coefficient matrix of $\mathscr{G}$ as a quadratic form of $\bar{X}$ is in the form $\left({ }^{*{d+1}}\right)$, the min-rank attack is also available on HFE and its complexity is $\left({ }_{d+2}^{n+d+2}\right)^{w} \ll n^{(d+2) w}[11,64]$.

From these facts, we see that both the decryption speed and the security of HFE are exponential of $d$, namely HFE has a serious trade-off between efficiency and security. Thus HFE itself has been considered to be impractical. In Sect. 4.2.2, we describe arrangements of MI and HFE to enhance the security.

# 数学建模代写

## 数学代写|数学建模代写math modelling代考|Extension Field Type

MI 的中心图 (第 $2.1$ 节，[69]) 由扩展域上的单变量单项式构造。虽然 MI 已经被打破，但产生的想法 $G$ 在一个扩展字段上用于多个 MPKC。中央地图 $G: k^{n} \rightarrow k^{m}$ 这种 MPKC 的一般描述如下。

$$G: k^{n} \stackrel{\phi_{N}}{\rightarrow} K^{N} \stackrel{\mathscr{g}}{\rightarrow} K^{M} \stackrel{\phi_{M}}{\longrightarrow} k^{m} .$$

$$\mathscr{G} i(\phi N(S(x)))^{q^{j}}=X^{t}\left(\Theta_{N} S \Theta_{N}^{-1}\right)^{t} G_{i}^{\left(q^{j}\right)}\left(\Theta_{N} S \Theta_{N}^{-1}\right) X+(\text { linear form of } X)$$

## 数学代写|数学建模代写math modelling代考|Hidden Field Equation

$$\mathscr{G}(X)=\sum_{0 \leq i \leq j \leq d} \alpha_{i j} X^{q^{i}+q^{j}}+\sum_{0 \leq i \leq d} \beta_{i} X^{q^{i}}+\gamma,$$

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