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    "# CS 6290: Privacy-enhancing Technologies\n",
    "## Assignment 1\n",
    "\n",
    "(Acknowledgement: this assignment is adapted from [Homework 9 for UVM CS 3990, developed by Joseph P. Near](https://github.com/jnear/cs3990-secure-computation/blob/main/homework/CS3990_HW_9.ipynb).)"
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   "source": [
    "import numpy as np\n",
    "import random\n",
    "import hashlib"
   ]
  },
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   "source": [
    "## The Petersen Graph\n",
    "\n",
    "[The Petersen Graph](https://en.wikipedia.org/wiki/Petersen_graph) has 10 vertices and 15 edges, and can be colored with 3 colors.\n",
    "\n",
    "![alt text](https://upload.wikimedia.org/wikipedia/commons/thumb/9/90/Petersen_graph_3-coloring.svg/220px-Petersen_graph_3-coloring.svg.png \"Title\")"
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    "coloring = {\n",
    "    # outer five nodes, clockwise from top\n",
    "    0: 'red',\n",
    "    1: 'blue', \n",
    "    2: 'green',\n",
    "    3: 'red',\n",
    "    4: 'blue',\n",
    "    # inner five nodes, clockwise from top\n",
    "    5: 'blue',\n",
    "    6: 'red',\n",
    "    7: 'red',\n",
    "    8: 'green',\n",
    "    9: 'green'\n",
    "}\n",
    "print('Number of nodes:', len(coloring))"
   ]
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   "source": [
    "edges = [\n",
    "    # outer shape, clockwise from top\n",
    "    (0, 1),\n",
    "    (1, 2),\n",
    "    (2, 3),\n",
    "    (3, 4),\n",
    "    (4, 0),\n",
    "    # inner shape, clockwise from top\n",
    "    (5, 0), (5, 7),\n",
    "    (6, 1), (6, 8),\n",
    "    (7, 2), (7, 9),\n",
    "    (8, 3), (8, 5),\n",
    "    (9, 4), (9, 6)\n",
    "]\n",
    "print('Number of edges:', len(edges))"
   ]
  },
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   "source": [
    "## Background\n",
    "\n",
    "**Your goal in this assignment** is to implement a zero-knowledge interactive proof protocol that allows the Prover to convince the Verifier that the Prover knows a valid 3-coloring for the Petersen graph - without revealing the coloring.\n",
    "\n",
    "You can find the description in [Matt Green's blog post](https://blog.cryptographyengineering.com/2014/11/27/zero-knowledge-proofs-illustrated-primer/).\n",
    "\n",
    "For those interested in a more theoretical understanding of Zero-Knowledge Proofs and the 3-coloring problem, we recommend exploring these learning materials:\n",
    "- **Interactive Zero-Knowledge Proofs:** Stanford University, CS355 Course Notes, [https://crypto.stanford.edu/cs355/18sp/lec3.pdf](https://crypto.stanford.edu/cs355/18sp/lec3.pdf)\n",
    "- **3-Coloring Zero-Knowledge Proof:** University of Illinois at Urbana-Champaign, CS498 Course Notes (Section 15.1), [https://courses.grainger.illinois.edu/cs498ac3/fa2020/Files/Lecture_15_Scribe.pdf](https://courses.grainger.illinois.edu/cs498ac3/fa2020/Files/Lecture_15_Scribe.pdf)"
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   "source": [
    "## Protocol\n",
    "\n",
    "The Prover and Verifier perform the following steps $n^2$ times, where $n$ is the number of vertices in the graph:\n",
    "\n",
    "- **Step 1: shuffle and commit.** The Prover randomizes the colors and commits to the coloring. The Prover sends the commitment to the Verifier.\n",
    "- **Step 2: challenge.** The Verifier picks a random edge in the graph.\n",
    "- **Step 3: response.** The Prover opens the commitment for the two vertices connected by the chosen edge. If the two colors are the same, the Verifier rejects."
   ]
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   "source": [
    "## Question 1 (7 points)\n",
    "\n",
    "Implement the above protocol for an interactive zero-knowledge proof of the graph coloring. (**6 points**)"
   ]
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    "class Prover:\n",
    "    def shuffle_and_commit(self, V):\n",
    "        # YOUR CODE HERE\n",
    "        raise NotImplementedError()\n",
    "\n",
    "        # send the commitment to the Verifier\n",
    "        V.receive_commitment(commitment)\n",
    "    \n",
    "    def response(self, edge, V):\n",
    "        # YOUR CODE HERE\n",
    "        raise NotImplementedError()\n",
    "        \n",
    "        # ask the verifier to check the response\n",
    "        V.check(c1, c2)\n",
    "\n",
    "class Verifier:\n",
    "    def receive_commitment(self, commitment):\n",
    "        # save the commitment for later\n",
    "        self.commitment = commitment\n",
    "    \n",
    "    def challenge(self, P):\n",
    "        # YOUR CODE HERE\n",
    "        raise NotImplementedError()\n",
    "\n",
    "        P.response(edge, self)\n",
    "\n",
    "    def check(self, color1, color2):\n",
    "        # YOUR CODE HERE\n",
    "        raise NotImplementedError()\n",
    "\n",
    "def run_protocol():\n",
    "    P = Prover()\n",
    "    V = Verifier()\n",
    "    for _ in range(15**2):\n",
    "        P.shuffle_and_commit(V)\n",
    "        V.challenge(P)\n",
    "    \n",
    "run_protocol()"
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   "source": [
    "**Note:** You should modify `edges` to demonstrate the Verifier will reject the proof with an invalid coloring. (**1 point**)."
   ]
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   "source": [
    "# YOUR CODE HERE\n"
   ]
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   "source": [
    "## Question 2 (1 points)\n",
    "\n",
    "In 2-5 sentences, argue that the protocol is zero-knowledge (it doesn't reveal anything about the witness)."
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    "you answer"
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   "source": [
    "## Question 3 (2 points)\n",
    "\n",
    "What is the probability that a cheating Prover is *not* caught in this protocol, and why? (You should give the proof of your answer)"
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