Error Detection and Correction: Nim; Secure Communication; RAID - - PowerPoint PPT Presentation

Error Detection and Correction: Nim; Secure Communication; RAID

Greg Plaxton Theory in Programming Practice, Fall 2005 Department of Computer Science University of Texas at Austin

The Game of Nim

• Initially, there are a number of piles of chips
• Two players move alternately
• A legal move consists of removing a positive number of chips from

some pile

• The first player who cannot make a legal move loses
• Thus, the player who removes the last chip wins

Theory in Programming Practice, Plaxton, Fall 2005

A Winning Strategy for Nim?

• A state is losing if the player to move next can be forced to lose:

– Examples: (1) The state in which all piles are empty; (2) Any state with an even number of nonempty piles, each of which has exactly

• ne chip; (3) Any state with two piles of equal size
• A state is winning if the player to move next can force a win

– Examples: (1) Any state with exactly one nonempty pile; (2) Any state with an odd number of nonempty piles, each of which has exactly one chip; (3) Any state with two piles of unequal size

• Is every state either losing or winning or could there be “uncertain”

states?

Theory in Programming Practice, Plaxton, Fall 2005

Every State is Either Losing or Winning

• Proof by strong induction on the total number of chips
• Base case: If there are no chips, the state is losing
• Induction hypothesis: Let k be a nonnegative integer and assume that

every state with a total of at most k chips is either losing or winning

• Induction step:

– Let S be an arbitrary state with a total of k + 1 chips – By the IH, every state reachable via a legal move from S is either losing or winning – If every state reachable via a legal move from S is winning, then S is losing; otherwise, it is winning

Theory in Programming Practice, Plaxton, Fall 2005

Towards a Characterization of Losing/Winning States

• Earlier we saw that if every pile has 0 or 1 chips, then the parity of

the number of nonempty piles determines whether a state is losing or winning

• Conjecture: A state is losing iff, for all nonnegative integers i, an even

number of piles have the property that the binary representation of the number of chips in the pile has a 1 in bit position i

• What is a more succinct statement of this conjecture?

Theory in Programming Practice, Plaxton, Fall 2005

Bad and Good States

• For any state S, let f(S) denote the XOR of the pile sizes
• A state S is bad if f(S) = 0, and good otherwise
• Theorem: A state is losing (resp., winning) iff it is bad (resp., good)
• In order to prove this claim, we will establish two lemmas

– Lemma 1: Any legal move executed in a bad state yields a good state – Lemma 2: For any good state, there exists a legal move to a bad state

• Why does the theorem follow?

Theory in Programming Practice, Plaxton, Fall 2005

Proof of the Theorem (using Lemmas 1 and 2)

• Proof by strong induction on the total number of chips
• Base case: If there are no chips, the state is losing and bad
• Induction hypothesis: Let k be a nonnegative integer and assume that

every bad (resp., good) state with at most k chips is losing (resp., winning)

• Induction step: Let state S have a total of k + 1 chips

– If S is bad, then by Lemma 1, every state reachable via a legal move from S is good, and by the IH, winning; hence, S is losing – If S is good, then by Lemma 2, there is a legal move from S to a state that is bad, and by the IH, losing; hence, S is winning

Theory in Programming Practice, Plaxton, Fall 2005

Proof of Lemma 1

• Lemma 1: Any legal move executed in a bad state yields a good state
• Suppose we are in a bad state S and we move to a state S′ by reducing

the number of chips in some pile from x to y

• Let i be a bit position where x has a 1 and y has a 0

– Such an i exists since x > y

• Note that bit position i of f(S′) is 1, so S′ is good

Theory in Programming Practice, Plaxton, Fall 2005

Proof of Lemma 2

• Lemma 2: For any good state, there exists a legal move to a bad state
• Let S be an arbitrary good state
• Let i be the highest bit position of f(S) containing a 1
• Let A be an arbitrary pile for which the number of chips, denoted by

x, has a 1 in bit position i of its binary representation – Such a pile A exists since bit position i of f(S) is 1

• Note that x ⊕ f(S) < x and (x ⊕ f(S)) ⊕ (x ⊕ f(S)) = 0, so we can

reach a bad state by reducing the number of chips in A from x to x ⊕ f(S)

Theory in Programming Practice, Plaxton, Fall 2005

Secure Communication: A Simple Approach

• Suppose a sender wishes to securely send a block of data x to a receiver
• ver an insecure communication channel
• Let us call a block of data random if each bit is independently set to 0

(resp., 1) with probability 1

2

• Suppose the sender and receiver share a secret random block

(sometimes called a key) k

• The sender sends y = x ⊕ k over the channel
• Note that the block y is random!

– Therefore an eavesdropper gains no information whatsoever about the message x from observing y

Theory in Programming Practice, Plaxton, Fall 2005

Disadvantages of this Approach

• Key exchange problem:

The sender and receiver need to agree in advance on the random block k, which must be sent over a secure channel

• The key should not be reused

– For example, suppose the eavesdropper sees x1 ⊕ k and x2 ⊕ k – The eavesdropper can calculate x1 ⊕ x2, which is (presumably) not random and might yield statistical clues towards partially deciphering the content of x1 and x2 – Furthermore, an eavesdropper who knows the content of a single block can determine k and thereby decipher all blocks

• Later in the course we will see a much better solution to the problem
• f secure communication

Theory in Programming Practice, Plaxton, Fall 2005

Fault-Tolerant Storage

• Suppose the records of a large corporate database are to be stored on

a collections of disks

• Occasionally one of the disks fails
• We would like to store the data in such a way that the entire content
• f the database can be recovered whenever any single disk fails
• Naive solution:

– Store two copies of each record (on different disks) – This doubles the storage requirement

• A better solution?

Theory in Programming Practice, Plaxton, Fall 2005

Redundant Array of Independent Disks (RAID)

• Store the database nonredundantly on k disks
• Use one additional disk to store the XOR of these k disks

– Here we are viewing the content of each disk as a block of bits

• If any disk fails, its content can be recovered since it is the XOR of the

remaining k disks

Theory in Programming Practice, Plaxton, Fall 2005