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我们提供的网络安全network security及其相关学科的代写，服务范围广, 其中包括但不限于:
非线性方法 nonlinear method functional analysis
变分法 Calculus of Variations
数学代写|网络安全作业代写network security代考|Pseudorandom Functions and Permutations
Pseudorandom functions (PRFs) generalize the notion of pseudorandom generators. Now, instead of considering “random-looking” strings we consider “random-looking” functions. As in our earlier discussion of pseudorandomness, it does not make much sense to say that any fixed function $f:{0,1}^{} \rightarrow{0,1}^{}$ is pseudorandom (in the same way it makes little sense to say that any fixed function is random). Instead, we must consider the pseudorandomness of a distribution on functions. Such a distribution is induced naturally by considering keyed functions, defined next.
A keyed function $F:{0,1}^{} \times{0,1}^{} \rightarrow{0,1}^{*}$ is a two-input function, where the first input is called the key and typically denoted by $k$. We say $F$ is efficient if there is a polynomial-time algorithm that computes $F(k, x)$ given $k$ and $x$. (We will only be interested in efficient keyed functions.) The security parameter $n$ dictates the key length, input length, and output length. That is, we associate with $F$ three functions $\ell_{\text {key }}, \ell_{\text {in }}$, and $\ell_{\text {out }}$; for any key $k \in{0,1}^{\ell_{k e y}(n)}$, the function $F_{k}$ is only defined for inputs $x \in{0,1}^{\ell_{i n}(n)}$, in which case $F_{k}(x) \in{0,1}^{\ell_{\text {out }}(n)}$. Unless stated otherwise, we assume for simplicity that $F$ is length preserving, meaning $\ell_{\text {key }}(n)=\ell_{\text {in }}(n)=\ell_{\text {out }}(n)=n$. (Note, however, that this is only to reduce notational clutter, and it is not uncommon to have pseudorandom functions that are not length preserving.) Let Func ${ }{n}$ denote the set of all functions mapping $n$-bit strings to $n$-bit strings. In typical usage a key $k \in{0,1}^{n}$ is chosen and fixed, and we are then interested in the single-input function $F{k}:{0,1}^{n} \rightarrow{0,1}^{n}$ defined by $F_{k}(x) \stackrel{\text { def }}{=} F(k, x)$ mapping $n$-bit input strings to $n$-bit output strings. A keyed function $F$ thus induces a distribution on functions in Func , where $^{2}$ the distribution is given by choosing a uniform key $k \in{0,1}^{n}$ and then considering the resulting single-input function $F_{k}$. We call $F$ pseudorandom if the function $F_{k}$ (for a uniform key $k$ ) is indistinguishable from a function chosen uniformly at random from the set Func $n$ of all functions having the same domain and range; that is, if no efficient adversary can distinguish-in a sense we more carefully define below-whether it is interacting with $F_{k}$ (for uniform $k$ ) or $f$ (where $f$ is chosen uniformly from Func ${ }_{n}$ ).
数学代写|网络安全作业代写network security代考|CPA-Security from a Pseudorandom Function
We focus here on constructing a CPA-secure fixed-length encryption scheme. By what we have said at the end of Section 3.4.3, this implies the existence of a CPA-secure encryption scheme for arbitrary-length messages. In Section $3.6$ we will discuss more efficient ways of encrypting messages of arbitrary length.
A naive attempt at constructing an encryption scheme from a pseudorandom permutation is to define $\operatorname{Enc}{k}(m)=F{k}(m)$. Although we expect that this “reveals no information about $m$ ” (since, if $f$ is a uniform permutation, then $f(m)$ is a uniform $n$-bit string for any $m)$, this method of encryption is deterministic and so cannot possibly be CPA-secure since encrypting the same plaintext twice will yield the same ciphertext.
Our CPA-secure construction uses randomized encryption. Specifically, we encrypt by applying a pseudorandom function to a random value $r \in{0,1}^{n}$ and XORing the output with the plaintext; the ciphertext includes both the result as well as $r$ (to enable the receiver to decrypt). See Figure $3.3$ and Construction 3.28. Encryption can again be viewed as XORing a pseudorandom pad with the plaintext (just like in the “pseudo-“one-time pad), with the major difference being the fact that here a fresh pseudorandom pad-that depends on $r$-is used each time a message is encrypted. (The pseudorandom pad is only “fresh” if the pseudorandom function is applied to a “fresh” value $r$ on which it has never been evaluated before. The proof below shows that with overwhelming probability this is always the case.)
网络安全作业代写
数学代写|网络安全作业代写NETWORK SECURITY代考|PSEUDORANDOM FUNCTIONS AND PERMUTATIONS
伪随机函数磷RFs概括伪随机生成器的概念。现在，我们不考虑“随机外观”的字符串，而是考虑“随机外观”的函数。正如我们之前对伪随机性的讨论，说任何固定函数 $f:{0,1}^{ } \rightarrow{0,1}^{ }$ 是伪随机性没有多大意义一世n吨H和s一种米和在一种是一世吨米一种ķ和sl一世吨吨l和s和ns和吨这s一种是吨H一种吨一种n是F一世X和dF在nC吨一世这n一世sr一种nd这米. 相反，我们必须考虑函数分布的伪随机性。这种分布是通过考虑接下来定义的键控函数自然得出的。
键控函数 $F:{0,1}^{} \times{0,1}^{} \rightarrow{0,1}^{*}$ is a two-input function, where the first input is called the key and typically denoted by $k$. We say $F$ is efficient if there is a polynomial-time algorithm that computes $F(k, x)$ given $k$ and $x$. (We will only be interested in efficient keyed functions.) The security parameter $n$ dictates the key length, input length, and output length. That is, we associate with $F$ three functions $\ell_{\text {key }}, \ell_{\text {in }}$, and $\ell_{\text {out }}$; for any key $k \in{0,1}^{\ell_{k e y}(n)}$, the function $F_{k}$ is only defined for inputs $x \in{0,1}^{\ell_{i n}(n)}$, in which case $F_{k}(x) \in{0,1}^{\ell_{\text {out }}(n)}$. Unless stated otherwise, we assume for simplicity that $F$ is length preserving, meaning $\ell_{\text {key }}(n)=\ell_{\text {in }}(n)=\ell_{\text {out }}(n)=n$. (Note, however, that this is only to reduce notational clutter, and it is not uncommon to have pseudorandom functions that are not length preserving.) Let Func ${ }{n}$ denote the set of all functions mapping $n$-bit strings to $n$-bit strings. In typical usage a key $k \in{0,1}^{n}$ is chosen and fixed, and we are then interested in the single-input function $F{k}:{0,1}^{n} \rightarrow{0,1}^{n}$ defined by $F_{k}(x) \stackrel{\text { def }}{=} F(k, x)$ mapping $n$-bit input strings to $n$-bit output strings. A keyed function $F$ thus induces a distribution on functions in Func , where $^{2}$ the distribution is given by choosing a uniform key $k \in{0,1}^{n}$ and then considering the resulting single-input function $F_{k}$. We call $F$ pseudorandom if the function $F_{k}$ (for a uniform key $k$ ) is indistinguishable from a function chosen uniformly at random from the set Func $n$ of all functions having the same domain and range; that is, if no efficient adversary can distinguish-in a sense we more carefully define below-whether it is interacting with $F_{k}$ (for uniform $k$ ) or $f$ (where $f$ is chosen uniformly from Func ${ }_{n}$ ).
数学代写|网络安全作业代写NETWORK SECURITY代考|CPA-SECURITY FROM A PSEUDORANDOM FUNCTION
我们在这里专注于构建一个 CPA 安全的固定长度加密方案。正如我们在第 3.4.3 节末尾所说的，这意味着存在用于任意长度消息的 CPA 安全加密方案。在部分3.6我们将讨论加密任意长度消息的更有效方法。
从伪随机排列构造加密方案的天真尝试是定义 $\operatorname{Enc} {k}米=F {k}米.一种l吨H这在GH在和和Xp和C吨吨H一种吨吨H一世s“r和在和一种lsn这一世nF这r米一种吨一世这n一种b这在吨米”(s一世nC和,一世FF一世s一种在n一世F这r米p和r米在吨一种吨一世这n,吨H和nF米一世s一种在n一世F这r米n−b一世吨s吨r一世nGF这r一种n是m)$，这种加密方法是确定性的，因此不可能是 CPA 安全的，因为两次加密相同的明文将产生相同的密文。
我们的 CPA 安全结构使用随机加密。具体来说，我们通过对随机值应用伪随机函数来加密r∈0,1n并将输出与明文进行异或；密文既包括结果，也包括r 吨这和n一种bl和吨H和r和C和一世在和r吨这d和Cr是p吨. 见图3.3和建筑 3.28。加密可以再次被视为对一个伪随机填充与明文进行异或运算j在s吨l一世ķ和一世n吨H和“ps和在d这−“这n和−吨一世米和p一种d, 主要区别在于这里有一个新的伪随机垫——这取决于r- 每次加密消息时使用。$r$-is used each time a message is encrypted. (The pseudorandom pad is only “fresh” if the pseudorandom function is applied to a “fresh” value $r$ on which it has never been evaluated before. The proof below shows that with overwhelming probability this is always the case.)
数学代写|网络安全作业代写network security代考 请认准UprivateTA™. UprivateTA™为您的留学生涯保驾护航。
