% Homework 10: Public key crypto review
\newcommand{\zo}{\{0,1\}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\getsr}{\leftarrow_R\;}
\newcommand{\Gp}{\mathbb{G}}
\newcommand{\iprod}[1]{\langle #1 \rangle}
\newcommand{\Epubcca}{E^{pub,cca}}
\newcommand{\Epubcpa}{E^{pub,cpa}}
\newcommand{\Epriv}{E^{priv,cca}}
\newcommand{\Sign}{S}
\newcommand{\Ver}{V}
\newcommand{\floor}[1]{\lfloor #1 \rfloor}
\newcommand{\ceil}[1]{\lceil #1 \rceil}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\onand}{\overline{\wedge}}
### Total of 120 points
(Most of this exercise is a review exercise on some of the notions we have encountered before.)
1. (25 points) Suppose that there exists an efficient algorithm $A$ that on input $m$ and $a \in \Z^*_m$ outputs the smallest number $r$ such that $a^r = 1 (\mod m)$. Prove that under this assumption there is an efficient (probabilistic) algorithm $B$ that on input $m=pq$ with $q (\mod 4) = p (\mod 4)=3$, outputs $p$ and $q$. You can follow the outline of the lecture notes, or see the footnote for hint on another approach[^hint]
[^hint]: For starters, you can assume for partial credit the following claim: with probability at least $1/100$, if we pick a random $a\in \Z^*_m$ then $a$ will have an even order and $a^{r/2} \neq -1 (\mod m)$. Using the claim you can reduce factoring to order finding in a similar way to how we reduced factoring to finding square roots. For full credit, prove the claim by first proving using the chinese remainder theorem that for every $a$, the order of $a$ modulo $m$ is the least common multiple of the order of $a$ modulo $P$ and the order of $a$ modulo $q$, and then use the fact that for every group $G$, if $G' \neq G$ is a subgroup of $G$ then $|G|/|G'| \geq 2$.
2. (50 points) Consider the following proof system for Alice to prove to Bob that a graph is 3 colorable:
* __Common input:__ Graph $G=(V,E)$ on $n$ vertices.
* __Alice (Prover) private input:__ A function $f:V\rightarrow \{1,2,3\}$ such that $f(i)\neq f(j)$ for every $\{i,j\}\in E$.
* __Step 1: Alice <- Bob:__ Bob selects $z,z' \getsr \zo^{10n}$ and sends $z,z'$ to Alice.
* __Step 2: Alice -> Bob:__ Alice selects $\pi$ to be a random permutation over $\{1,2,3\}$ and defines the functions $f':V\rightarrow \{1,2,3\}$ as $f'(i)=\pi(f(i))$. For $i=1..n$, Alice chooses $w_i \getsr \zo^n$ and sends to Bob $y_i = PRG(w_i)+f'(i)z +(f'(i) \mod\; 3)z' (\mod 2)$ where $PRG:\zo^n\rightarrow\zo^{10n}$ is a pseudorandom generator and vector addition and vector/scalar multiplication are defined as usual.
* __Step 3: Bob <- Alice:__ Bob selects a random edge $\{i,j \} \in E$ and sends $i$ and $j$ to Alice.
* __Step 4: Alice -> Bob:__ Alice checks that $\{i,j\}\in E$ (otherwise she aborts) and if so sends the strings $w_i,w_j$ and the values $f'(i),f'(j)$.
* __Bob's decision:__ Bob accepts the proof iff $f'(i),f'(j)$ as sent by Alice are two distinct numbers in $\{1,2,3\}$ and the strings she sent satisfy the equations $y_i = PRG(w_i)+f'(i)z +(f'(i) \mod\; 3)z' (\mod 2)$ and $y_j = PRG(w_j)+f'(j)z +(f'(j) \mod\; 3)z' (\mod 2)$
Prove that this system is a zero knowledge proof system for the 3 coloring problem by showing the following:
a. (Completeness, 10 points): Prove that if Alice and Bob are given inputs as above and both follow the protocol then Bob will accept the proof with probability $1$.
b. (Soundness, 15 points): Prove that if there exists no 3-coloring for $G$ (i.e., for every coloring of $G$'s vertices in $\{1,2,3\}$ there is some edge $\{i,j\}$ such that both $i$ and $j$ receive the same color), then with probability at least $1/(10n^2)$ Bob will reject the proof. (This probability can be amplified to more than $1-2^{-k}$ by $100kn^2$ repetitions).
c. (Zero knowledge, 25 points) Prove that for every polynomial-time strategy $B^*$ used by Bob, there exists an efficient algorithm $S^*$, so that for every 3-colorable graph $G$ and coloring $f$, the output of $S^*(G)$ is computationally indistinguishabl from the transcript $B^*$ observes after interacting with the honest strategy of Alice on public input $G$ and private input $x$. (For partial credit of 15 points, prove only _honest verifier zero knowledge_ : that the above holds when $B^*$ is the honest strategy of Bob.)
3. KL 11.17 (20 points)
4. KL 12.14 (10 points)
5. KL 13.17 (15 points)