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Problem 13

Let \(M O D E X P=\\{\langle a, b, c, p\rangle \mid a, b, c\), and \(p\) are positive binary integers such that \(\left.a^{b} \equiv c(\bmod p)\right\\}\). Show that \(M O D E X P \in P\). (Note that the most obvious algorithm doesn't run in polynomial time. Hint: Try it first where \(b\) is a power of 2 .)

Expert verified

The problem \(MODEXP\), defined as \(MODEXP =\\{\langle a, b, c, p\rangle \mid a, b, c, and p\) are binary integers such that \(a^{b} \equiv c(\bmod p)\right\\}\), can be solved using an algorithm running in polynomial time. In the case where \(b\) is a power of 2, the result can be efficiently computed by successively squaring the base mod \(p\). This concept can be extended to arbitrary exponents \(b\) with a divide-and-conquer approach, breaking down the exponentiation into smaller components and utilizing the squaring method. This approach, known as "exponentiation by squaring", evaluates even and odd exponents differently: for even exponents, we compute \((a^{(b/2)} \cdot a^{(b/2)}) \pmod{p}\), and for odd exponents, we compute \((a^{(b-1)} \cdot a) \pmod{p}\). This algorithm shows that \(MODEXP\) is indeed a class \(P\) problem, thus, the problem \(MODEXP \in P\), as required.

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Chapter 7

In a directed graph, the indegree of a node is the number of incoming edges and the outdegree is the number of outgoing edges. Show that the following problem is NP-complete. Given an undirected graph \(G\) and a designated subset \(C\) of \(G\) 's nodes, is it possible to convert \(G\) to a directed graph by assigning directions to each of its edges so that every node in \(C\) has indegree 0 or outdegree 0 , and every other node in \(G\) has indegree at least 1 ?

Chapter 7

A subset of the nodes of a graph \(G\) is a dominating set if every other node of \(G\) is adjacent to some node in the subset. Let DOMINATING-SET $=\\{\langle G, k\rangle \mid G\( has a dominating set with \)k\( nodes \)\\} .$ Show that it is NP-complete by giving a reduction from VERTEX-COVER.

Chapter 7

Call a regular expression star-free if it does not contain any star operations. Then, let $E Q_{\mathrm{SF}-\mathrm{REX}}=\\{\langle R, S\rangle \mid R\( and \)S\( are equivalent star-free regular expressions \)\\}$. Show that \(E Q_{\mathrm{SF}-\mathrm{REX}}\) is in coNP. Why does your argument fail for general regular expressions?

Chapter 7

Call a regular expression star-free if it does not contain any star operations. Then, let $E Q_{\mathrm{SF}-\mathrm{REX}}=\\{\langle R, S\rangle \mid R\( and \)S\( are equivalent star-free regular expressions \)\\}$. Show that \(E Q_{\mathrm{SF}-\mathrm{REX}}\) is in coNP. Why does your argument fail for general regular expressions?

Chapter 7

Let \(D O U B L E-S A T=\\{\langle\phi\rangle \mid \phi\) has at least two satisfying assignments \(\\}\). Show that DOUBLE-SAT is NP-complete.

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