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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 Generators
A pseudorandom generator $G$ is an efficient, deterministic algorithm for transforming a short, uniform string (called a seed) into a longer, “uniformlooking” (or “pseudorandom”) output string. Stated differently, a pseudorandom generator uses a small amount of true randomness in order to generate a large amount of pseudorandomness. This is useful whenever a large number of random(-looking) bits are needed, since generating true random bits is often difficult and slow. (See the discussion at the beginning of Chapter 2.) Pseudorandom generators have been studied since at least the 1940 s when they were used for running statistical simulations. In that context, researchers proposed various statistical tests that a pseudorandom generator should pass in order to be considered “good.” As a simple example, one could require that the first bit of the output of a pseudorandom generator should be equal to 1 with probability very close to $1 / 2$ (where the probability is taken over uniform choice of the seed), since the first bit of a uniform string is equal to 1 with probability exactly $1 / 2$. As another example, the parity of any fixed subset of the output bits should also be 1 with probability very close to $1 / 2$. More complex statistical tests can also be considered.
This historical approach to determining the quality of some candidate pseudorandom generator is unsatisfying, as it is not clear when passing some set of statistical tests is sufficient to guarantee the soundness of using a candidate pseudorandom generator for some application. (In particular, there may be another statistical test that does successfully distinguish the output of the generator from true random bits.) The historical approach is even more problematic when using pseudorandom generators for cryptographic applications; there, security may be compromised if any attacker is able to distinguish the output of a pseudorandom generator from uniform, and we do not know in advance what strategy an attacker might use.
数学代写|网络安全作业代写network security代考|Proofs by Reduction
If we wish to prove that a given construction (e.g., encryption scheme) is computationally secure, then-unless the scheme is information-theoretically secure – we must rely on unproven assumptions. Our strategy will be to assume that some mathematical problem is hard, or that some low-level cryptographic primitive is secure, and then to prove that the given construction based on that problem/primitive is secure as long as our initial assumption is correct. In Section 1.4.2 we have already explained in great detail the advantages of this approach, so we do not repeat those arguments here.
A proof that some cryptographic construction $\Pi$ is secure as long as some underlying problem $X$ is hard generally proceeds by presenting an explicit reduction showing how to transform any efficient adversary $\mathcal{A}$ that succeeds in “breaking” $\Pi$ into an efficient algorithm $\mathcal{A}^{\prime}$ that solves X. Since this is so important, we walk through a high-level outline of the steps of such a proof in detail. (We will see numerous concrete examples throughout the book, beginning with the proof of Theorem $3.16$ in the next section.) We start with the assumption that some problem $X$ cannot be solved (in some precisely defined sense) by any polynomial-time algorithm, except with negligible probability. We then want to prove that some cryptographic construction $\Pi$ is secure (again, in some sense that is precisely defined). A proof by reduction proceeds via the following steps (see also Figure 3.1):
- Fix some efficient (i.e., probabilistic polynomial-time) adversary $\mathcal{A}$ attacking $\Pi$. Denote this adversary’s success probability by $\varepsilon(n)$.
- Construct an efficient algorithm $\mathcal{A}^{\prime}$ that attempts to solve problem $\mathrm{X}$ by using adversary $\mathcal{A}$ as a subroutine. An important point here is that $\mathcal{A}^{\prime}$ knows nothing about how $\mathcal{A}$ works; the only thing $\mathcal{A}^{\prime}$ knows is that $\mathcal{A}$ is expecting to attack $\Pi$. So, given some input instance $x$ of problem $X$, our algorithm $\mathcal{A}^{\prime}$ will simulate for $\mathcal{A}$ an instance of $\Pi$ such that:
(a) As far as $\mathcal{A}$ can tell, it is interacting with $\Pi$. That is, the view of $\mathcal{A}$ when run as a subroutine by $\mathcal{A}^{\prime}$ should be distributed identically to (or at least close to) the view of $\mathcal{A}$ when it interacts with $\Pi$ itself.
(b) When $\mathcal{A}$ succeeds in “breaking” the instance of $\Pi$ that is being simulated by $\mathcal{A}^{\prime}$, this should allow $\mathcal{A}^{\prime}$ to solve the instance $\mathrm{x}$ it was given, at least with inverse polynomial probability $1 / p(n)$.
数学代写|网络安全作业代写NETWORK SECURITY代考|EAV-Security from a Pseudorandom Generator
A pseudorandom generator provides a natural way to construct a secure, fixed-length encryption scheme with a key shorter than the message. Recall that in the one-time pad (see Section 2.2), encryption is done by XORing a random pad with the message. The crucial insight is that we can use a pseudorandom pad instead. Rather than sharing this long, pseudorandom pad, however, the sender and receiver can instead share a uniform seed that is used to generate the pad when needed (see Figure $3.2$ ); this seed will be shorter than the pad and hence shorter than the message. As for security, the intuition is that a pseudorandom string “looks random” to any polynomial-time adversary and so a computationally bounded eavesdropper cannot distinguish between a message encrypted using the one-time pad or a message encrypted using this “pseudo-“one-time pad encryption scheme.
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 GENERATORS
一个伪随机生成器G是一种有效的确定性算法，用于转换短的、统一的字符串C一种ll和d一种s和和d变成一个更长的，“统一的”这r“ps和在d这r一种nd这米”输出字符串。换句话说，伪随机生成器使用少量的真实随机性来生成大量的伪随机性。这在大量随机数时很有用−l这这ķ一世nG位是必需的，因为生成真正的随机位通常既困难又缓慢。小号和和吨H和d一世sC在ss一世这n一种吨吨H和b和G一世nn一世nG这FCH一种p吨和r2.至少从 1940 年代开始就已经研究了伪随机发生器，当时它们被用于运行统计模拟。在这种情况下，研究人员提出了伪随机生成器应该通过的各种统计测试，以便被认为是“好的”。作为一个简单的例子，可以要求伪随机发生器的输出的第一位应该等于 1，概率非常接近1/2 在H和r和吨H和pr这b一种b一世l一世吨是一世s吨一种ķ和n这在和r在n一世F这r米CH这一世C和这F吨H和s和和d, 因为统一字符串的第一位等于 1 的概率恰好是1/2. 作为另一个例子，输出比特的任何固定子集的奇偶校验也应该是 1，概率非常接近1/2. 也可以考虑更复杂的统计测试。
这种确定某些候选伪随机生成器质量的历史方法并不令人满意，因为尚不清楚何时通过某些统计测试足以保证将候选伪随机生成器用于某些应用程序的可靠性。一世np一种r吨一世C在l一种r,吨H和r和米一种是b和一种n这吨H和rs吨一种吨一世s吨一世C一种l吨和s吨吨H一种吨d这和ss在CC和ssF在ll是d一世s吨一世nG在一世sH吨H和这在吨p在吨这F吨H和G和n和r一种吨这rFr这米吨r在和r一种nd这米b一世吨s.在将伪随机生成器用于加密应用时，历史方法的问题会更大。在那里，如果任何攻击者能够将伪随机生成器的输出与均匀区分开来，那么安全性可能会受到损害，而且我们事先不知道攻击者可能使用什么策略。
数学代写|网络安全作业代写NETWORK SECURITY代考|PROOFS BY REDUCTION
如果我们想证明一个给定的结构和.G.,和nCr是p吨一世这nsCH和米和在计算上是安全的，那么除非该方案在信息理论上是安全的，否则我们必须依赖未经证实的假设。我们的策略是假设一些数学问题是困难的，或者一些低级密码原语是安全的，然后证明基于该问题/原语的给定结构是安全的，只要我们的初始假设是正确的。在 1.4.2 节中我们已经非常详细地解释了这种方法的优点，所以我们在这里不再重复这些论点。
一些密码构造的证明圆周率只要存在一些潜在问题，它就是安全的X很难通过提出一个明确的减少来展示如何改变任何有效的对手一种成功“破”圆周率变成一个有效的算法一种′这解决了 X。由于这非常重要，我们将详细介绍此类证明的步骤的高级概述。在和在一世lls和和n在米和r这在sC这nCr和吨和和X一种米pl和s吨Hr这在GH这在吨吨H和b这这ķ,b和G一世nn一世nG在一世吨H吨H和pr这这F这F吨H和这r和米$3.16$一世n吨H和n和X吨s和C吨一世这n.我们从假设存在一些问题开始X无法解决一世ns这米和pr和C一世s和l是d和F一世n和ds和ns和通过任何多项式时间算法，除非概率可忽略不计。然后我们想证明一些密码构造圆周率是安全的一种G一种一世n,一世ns这米和s和ns和吨H一种吨一世spr和C一世s和l是d和F一世n和d. 通过以下步骤进行归约证明s和和一种ls这F一世G在r和3.1:
- 修复一些高效的一世.和.,pr这b一种b一世l一世s吨一世Cp这l是n这米一世一种l−吨一世米和对手一种进攻圆周率. 将这个对手的成功概率表示为e(n).
- 构造一个高效的算法一种′试图解决问题的X通过使用对手一种作为子程序。这里重要的一点是一种′一无所知一种作品; 唯一的事情一种′知道是一种期待攻击圆周率. 所以，给定一些输入实例X问题的X, 我们的算法一种′将模拟一种的一个实例圆周率这样：
一种据，直到…为止一种可以说，它正在与圆周率. 也就是说，观点一种当作为子程序运行时一种′应该相同地分配给这r一种吨l和一种s吨Cl这s和吨这的观点一种当它与圆周率本身。
b什么时候一种成功“打破”了圆周率正在模拟一种′，这应该允许一种′解决实例X它是给出的，至少具有逆多项式概率1/p(n).
数学代写|网络安全作业代写NETWORK SECURITY代考|EAV-SECURITY FROM A PSEUDORANDOM GENERATOR
伪随机生成器提供了一种自然的方式来构建安全的、固定长度的加密方案，其密钥比消息短。回想一下，在一次性垫s和和小号和C吨一世这n2.2, 加密是通过对消息进行异或随机填充来完成的。关键的见解是我们可以使用伪随机垫代替。然而，发送者和接收者可以共享一个统一的种子，而不是共享这个长长的伪随机填充物，该种子用于在需要时生成填充物s和和F一世G在r和$3.2$; 这个种子将比垫短，因此比消息短。至于安全性，直觉是伪随机字符串对于任何多项式时间的对手来说“看起来都是随机的”，因此计算受限的窃听者无法区分使用一次性密码加密的消息或使用这种“伪”加密的消息-time pad 加密方案。
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