## 数学代写|数学建模代写math modelling代考|MAT3104

2022年7月28日

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## 数学代写|数学建模代写math modelling代考|Noncommutative Version

In Sects. 3.2.1-4, we describe MPKCs whose central maps are derived from polynomial maps over extension fields. Such constructions can be generalized to rings, not necessarily fields. In fact, there have been several MPKCs constructed on noncommutative rings $[54,103,109,110,112]$. However, we cannot recommend such constructions strongly since the following theorem is well-known (see e.g. [6]).
The Artin-Wedderburn theorem. A ring $\mathscr{R}$ is a semi-simple if and only if there exist integers $n_{1}, \ldots, n_{l} \geq 1$ and division rings $K_{1}, \ldots, K_{l}$ such that
$$\mathscr{R} \simeq \mathrm{M}{n{1}}\left(K_{1}\right) \oplus \cdots \oplus \mathrm{M}{n{l}}\left(K_{l}\right),$$
where $\mathrm{M}{n}(K)$ is the ring of $n \times n$ matrices of $K$-entries. Furthermore, due to Wedderburn’s theorem, we see that, if a semi-simple ring $\mathscr{R}$ is finite, then the rings $K{1}, \ldots, K_{l}$ are commutative. For example, let
$$\mathscr{R}:=\left{a_{1} \sigma_{1}+\cdots+a_{5} \sigma_{5} \mid a_{1}, \ldots, a_{5} \in k\right}$$ $\sigma_{4}:=\left(\begin{array}{cc}1 & 1 \ 1\end{array}\right), \sigma_{5}:=\left({ }{1}^{1} \begin{array}{c}1 \ 1\end{array}\right)$. Define $\delta{1}, \ldots, \delta_{5} \in \mathscr{R}$ by
where $\alpha \in \mathscr{R}$ satisfies $\alpha \neq 1, \alpha^{3}=1$. It is easy to see that the elements $\delta_{1}, \ldots, \delta_{5}$ have the following multiplicative relations.

## 数学代写|数学建模代写math modelling代考|ABC Encryption Scheme

In the $A B C$ (or Simple Matrix) encryption scheme proposed by Tao et al. [99], the central map $G$ is generated by products among three matrices $A, B, C$. It is generalized as follows. Let $n, m \geq 1$ be integers with $m:=2 n, \mathscr{R}$ a ring over $k$ with $[\mathscr{R}: k]=n$ and $\left{\xi_{1}, \ldots, \xi_{n}\right} \subset \mathscr{R}$ is a basis of $\mathscr{R}$ over $k$. Denote by $\phi: k^{n} \rightarrow \mathscr{R}, \phi_{2}:$ $k^{m} \rightarrow \mathscr{R}^{2}$ one-to-one maps, e.g. $\phi\left(x_{1}, \ldots, x_{n}\right)=x_{1} \xi_{1}+\cdots+x_{n} \xi_{n}$ and $\phi_{2}\left(y_{1}, \ldots\right.$, $\left.y_{m}\right)=\left(y_{1} \xi_{1}+\cdots+y_{n} \xi_{n}, y_{n+1} \xi_{1}+\cdots+y_{m} \xi_{n}\right)$ for $x_{1}, \ldots, x_{n}, y_{1}, \ldots, y_{m} \in k$, and $\mathscr{B}, \mathscr{C}: k^{n} \rightarrow k^{n}$ linear maps. For $x \in k^{n}$, put $A=A(x):=\phi(x), B=B(x):=$ $\phi(\mathscr{B}(x)), C=C(x):=\phi(\mathscr{C}(x)), E_{1}=E_{1}(x):=A \cdot B, E_{2}=E_{2}(x):=A \cdot C$ and $E(x):=\left(E_{1}(x), E_{2}(x)\right)$. The central map $G: k^{n} \rightarrow k^{m}$ is defined by
$$G:=\phi_{2}^{-1} \circ E \circ \phi .$$
For $Y_{1}, Y_{2} \in \mathscr{R}$, one finds $x \in k^{n}$ with $E_{1}(x)=Y_{1}$ and $E_{2}(x)=Y_{2}$ by solving a system of linear equations derived from $C(x)=B(x) Y_{1}^{-1} Y_{2}$ or $B(x)=C(x) Y_{2}^{-1} Y_{1}$.
It is easy to see that the original $\mathrm{ABC}$ encryption scheme [99] is just same to the case that $\mathscr{R}=\mathrm{M}_{r}(k)$ with $r^{2}=n$, and the extension field cancelation (EFC) [97] is essentially expressed as an $\mathrm{ABC}$ encryption scheme in the case that $\mathscr{R}$ is an $n$ extension field of $k$.

The decryption of this scheme is simple and quite efficient. However, the decryption fails when $A$ is not invertible. Especially, the probability of decryption failure for the original $\mathrm{ABC}$ encryption scheme [99] is about $q^{-1}$, which is not negligible. To reduce the probability of decryption failure, several arrangements have been proposed, e.g., taking $q$ large, using rectangular matrices instead of $A, B, C$ et al. [100], using a tensor type matrix as $S$ [88]. However, the security for such arrangements should be studied carefully. It was shown that the tensor type $S$ is a weak key [56].
For the security, it is known that the min-rank attack and the linearization attack are available on this encryption scheme. For the original $\mathrm{ABC}$ [99], the compleximore, Moody et al. [73] proposed another attack on this scheme with the complexity $O\left(q^{r+4}\right.$. (polyn.)). Then this encryption scheme (presently) has a sub-exponential time security of $n$. For EFC, it is known that the linearization attack can recover plaintexts easily. To prevent it, the authors of [97] recommended to use the minus and the projection of EFC. In [39], the cubic version of $\mathrm{ABC}$ was proposed; the polynomials in $A$ are quadratic and then those in $F, G$ are cubic. Though the security against the direct attack is improved, the security against the linearization attack is almost same to the original $\mathrm{ABC}$

# 数学建模代写

## 数学代写|数学建模代写math modelling代考|Noncommutative Version

$[54,103,109,110,112]$. 但是，我们不能强烈推荐这种结构，因为以下定理是众所周知的 (参见例如 [6])。

Artin-Wedderburn 定理。戒指 $\mathscr{R}$ 是半简单的当且仅当存在整数 $n_{1}, \ldots, n_{l} \geq 1$ 和分割环 $K_{1}, \ldots, K_{l}$ 这样
$$\mathscr{R} \simeq \operatorname{Mn} 1\left(K_{1}\right) \oplus \cdots \oplus \operatorname{Mnl}\left(K_{l}\right)$$

$$\sigma_{4}:=\left(\begin{array}{ll} 1 & 11 \end{array}\right), \sigma_{5}:=\left(1^{1} 11\right) \text {. 定义 } \delta 1, \ldots, \delta_{5} \in \mathscr{R} \text { 在 }$$

## 数学代写|数学建模代写math modelling代考|ABC Encryption Scheme

$\phi: k^{n} \rightarrow \mathscr{R}, \phi_{2}: k^{m} \rightarrow \mathscr{R}^{2}$ 一对一的映射，例如
$\phi\left(x_{1}, \ldots, x_{n}\right)=x_{1} \xi_{1}+\cdots+x_{n} \xi_{n}$ 和 $\phi_{2}\left(y_{1}, \ldots\right.$
$\left.y_{m}\right)=\left(y_{1} \xi_{1}+\cdots+y_{n} \xi_{n}, y_{n+1} \xi_{1}+\cdots+y_{m} \xi_{n}\right)$ 为了
$x_{1}, \ldots, x_{n}, y_{1}, \ldots, y_{m} \in k$ ，和 $\mathscr{B}, \mathscr{C}: k^{n} \rightarrow k^{n}$ 线性地图。为了 $x \in k^{n}$ ，放 $A=A(x):=\phi(x), B=B(x):=$
$\phi(\mathscr{B}(x)), C=C(x):=\phi(\mathscr{C}(x)), E_{1}=E_{1}(x):=A \cdot B, E_{2}=E_{2}(x):=A \cdot C$ 和 $E(x):=\left(E_{1}(x), E_{2}(x)\right)$. 中央地图 $G: k^{n} \rightarrow k^{m}$ 定义为
$$G:=\phi_{2}^{-1} \circ E \circ \phi .$$

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## MATLAB代写

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