University of Virginia, Department of Computer Science CS588: Cryptology - Principles and Applications, Fall 2001

 Problem Set 5: Quantum Crypto Coloring Book Out: 14 November 2001 Due: 26 November 2001, before class

This problem set is optional. If you do it, it will replace your lowest grade on a previous problem set, or count as your score/2 bonus points (whichever helps you most). If you don't do it, you will be expected to produce an even more spectacular project than is already expected.

Collaboration Policy - Read carefully, changed from previous assignments

You are encouraged to work with other students on this problem set. You may work with anyone you want.

If you work with more than one other person, everyone must write up their answers independently, and understand completely everything you turn in.

If you work with only one other person, you may turn in a signle problem set with both of your names on it, and each student will receive the same grade.

Working together means discussing the questions and critiquing possible solutions; it does not permit splitting up questions in a group.

You may consult any outside resources you wish including books, papers, web sites and people. If you use resources other than the class materials, indicate what you used along with your answer.

Problem set answers may be hand-written, but only if your hand writting is neat enough for us to read it. For full credit, answers must be clear and concise.

### 1. Maury Bond's Coloring Crypto

This question is inspired by a question by Giuseppe Ateniese.

Maury Bond's secret agents (from Problem Set 1, question 6) never did manage to decode the message and locate the super ray gun. Last he heard, they were still in Borneo XOR'ing random bits.

Maury has decided to recruit three new secret agents: Abby Avaricous, Billy Badd and Carrie A. Grudge. This time, however, he will avoid the problems with revealing bits in order by encoding the mesesage on three transparencies such that they can be placed on top of each other to reveal the message all at once. Any two transparencies by themselves should reveal no information, but when the three are aligned correctly on top of each other, everyone will see the message appear at the same time.

a. (30) Explain how Maury can create the three transparencies. You may find it useful to use an encoding scheme based on the primary subtractive colors - cyan (C), magenta (M) and yellow (Y). Cyan absorbs red; magenta absorbs green; and yellow absorbs blue. The following relations hold when transparencies of these colors are placed on top of each other:

Yellow + Cyan = Green
Yellow + Magenta = Red
Cyan + Magenta = Blue
Cyan + Magenta + Yellow = Black

b. (15) Prove that your scheme is perfectly secure in an information-theoretic sense. That is, no two transparencies by themselves reveal anything interesting about the message.

c. (15) Maury fears that one of the three agents may be killed. He decides he is no longer worried about two agents conspiring. Instead, he would like to make sure that any two agents can combine their transparencies to reveal the message, but each agent by herself has no information. Describe a scheme that satisfies this requirement.

d. (up to 50 bonus points) Implement your scheme from either part a or c to produce transparencies encoding a message that demonstrate your scheme works.

### 2. Quantum Leap

Bennet's Quantum Key Distribution scheme from Lecture 16 allows Alice and Bob to establish a shared secret key with perfect secrecy with on average 2 photons transmitted per bit (i.e., for each photon transmitted, there is a 50% chance Alice and Bob will agree on a key bit).

Alice, Bob and Coleen would now like to establish a common secret key (all three of them will know the same key, but no one else can know anything).

(40) Describe a quantum key distribution scheme three people can use to establish a secret key. For full credit, your distribution scheme should require less than 4 photons transmitted per bit, and must be perfectly secure against all forms of both passive and active eavesdropping.

### 3. Coloring

(no credit, 20 bonus points if I use your drawings for future slides)
Draw Alice, Bob and Eve.