## 数学代写|复分析作业代写Complex function代考|MAST30021

2022年12月26日

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## 数学代写|复分析作业代写Complex function代考|Biholomorphic Cryptosystems

In classical cryptography, a cryptosystem is a five-tuple $(\mathcal{P}, \mathcal{C}, \mathcal{K}, \mathcal{E}, \mathcal{D})$, where the following conditions are satisfied. $\mathcal{P}$ is a finite set of possible plaintexts $\mathcal{C}$ is a finite set of possible ciphertexts $\mathcal{K}$, the keyspace, is a finite set of possible keys. For each $K \in \mathcal{K}$, there is an encryption rule $e_K \in \mathcal{E}$ and a corresponding decryption rule $d_K \in \mathcal{D}$. Each $e_K: \mathcal{P} \rightarrow \mathcal{C}$ and $d_K: \mathcal{C} \rightarrow \mathcal{P}$ are functions such that $d_K\left(e_K\right)(x)=$ $x$ for every plaintext element $x \in \mathcal{P}$.

The purpose of the present section is to give an extension of this definition which is based on theory of complex variables and to show the differences of this extension with the classical case. In this direction, we point out that translating such a classical definition to the context of complex variables induces a natural change in the keyspace, from the discrete state to an uncountable infinite structure. It follows that an encryption method which based on the theory of complex analysis is beyond the capacities of modern computers, even of future quantum computers. As a consequence, the rules in the context of theory of complex analysis can become so complicated, from the point of view of constructive approximations, that it becomes impossible to achieve decryption by using electronic machines or advanced computer technology.

After these brief and very general introductory remarks, we are able to go to the foundation of complex cryptosystems. To this end, we first give the following general definition. To do so, we may remark that, in practice, the capacity of each message may not exceed a certain number of characters, so we can assume that the length of each plaintext in $\mathcal{P}$ equals a given number, say $n$. Otherwise, you may add at the end of the plain text, the symbol of blank space, so many times so that the length of the resulting final plaintext which will result equals to $n$. Under this assumption, we are now in position to define biholomorphic cryptosystems.

Definition 5 Let us consider two domains $\Omega \subset \mathbb{C}^n$ and $D \subset \mathbb{C}^n$ which together constitute the encryption environment. A finite biholomorphic cryptosystem is a four-tuple $(\mathcal{P}, \mathcal{K}, \mathcal{E}, \mathcal{D})$, where the following conditions are satisfied.

## 数学代写|复分析作业代写Complex function代考|Dynamics of Biholomorphic Cryptosystems

The idea of considering biholomorphic cryptosystems is not new. As a concept, the biholomorphic cryptosystem is nested in the meaning of the chain of a sequence of biholomorphic mappings. Already, in 2005, Han Peters in his doctoral thesis examined the dynamic behaviour of the composition of a sequence of automorphisms, in the special case in which each mapping which participates in the composition has a single attracting fixed point. In this section, we discuss the generalization of the results of Han Peters.

Peters having as a springboard earlier work of Rudin and Rosay raised the following question. Let $f_0, f_1, \ldots$ be a sequence of automorphisms of a complex manifold all having a single attracting fixed point. Under what conditions is the basin of attraction biholomorphically equivalent to a complex Euclidean space? Here, by a basin of attraction, it is meant the set of $p$ points whose (nonautonomous) orbits converge to fixed point. This question was motivated by a question about stable manifolds. A stable manifold is a generalization of a basin of attraction to the case where there is not a fixed point. Peters proved a more general proposition.

Theorem 6 A stable manifold is always biholomorphic to complex Euclidean space if the following conjecture holds: Let $f_0, f_1 \ldots$ be a sequence of automorphisms of $\mathbb{C}^n$ that fix the origin. Assume that there exist $a, b \in \mathbb{R}$ with $0<a<b<1$ so that for any $z$ in the unit ball and any $k \in \mathbb{N}$ the following holds:

(C) $\quad a|z|<\left|f_k(z)\right|<b|z|$.
Then the basin of attraction of the sequence $f_0, f_1, \ldots$ is biholomorphic to $\mathbb{C}^n$.
Several examples show that a basin of attraction of a sequence of biholomorphic mappings is not biholomorphic to $\mathbb{C}^n$ unless some assumptions are made on the rate at which different orbits converge to the attracting fixed point. However, it is showed that given any sequence of automorphisms with a common attracting fixed point, the basin of attraction is biholomorphic to $\mathbb{C}^n$ if the mappings are repeated often enough.

In what follow we will discuss the extension of the results of Peters in the case of biholomorphic cryptosystems. To this end, without loss of generality and by expanding the interpretation of Definition 2, we can extend the concept of a (finite) holomorphic cryptosystem in the case of an infinite encryption chain.

# 复分析代写

## 数学代写|复分析作业代写Complex function代考|Dynamics of Biholomorphic Cryptosystems

(C) $\quad a|z|<\left|f_k(z)\right|<b|z|$.

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## MATLAB代写

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