Intro to Quantum Cryptography Algorithms

1
Intro to Quantum Cryptography Algorithms

• Andrew Hamel
• EECS 598 Quantum Computing
• FALL 2001

2
References

• Ekert, A. From quantum code-making to quantum
  code-breaking, 1997
• Brassard, et al. A Quantum Bit Commitment Scheme
  Provably Unbreakable by both Parties, 1993

3
Outline

• Classical Cryptography overview
• Problems with classical cryptography
• Quantum Algorithms
• Quantum Bit Commitment
• Quantum Key Distribution Introduction
• Conclusions

4
Classical Cryptography

• Simple Methods
• Transposition
• Arrange the plaintext in a special permutation
• Substitution
• Replace letters of the plaintext with other
  letters or symbols in a certain way
• Caesars Cipher
• COLD -gt FROG

5
Classical Cryptography

• Problem with simple methods
• Security depends on the secrecy of the entire
  encrypting/decrypting process
• Need a way to ensure secrecy even if the
  encryption process is compromised

6
Classical Cryptography

• Key-based Cryptography
• Private Key
• Secret key locks and unlocks data
• Encrypt EPri(P) C
• Decrypt DPri(C) P
• Public Key
• Separate keys to lock and unlock data
• Encrypt EPub(P) C
• Decrypt DPri(C) P

7
Classical Cryptography

• Problem with private-key encryption
• Depends entirely on the secrecy of the key
• Requires two parties who initially share no
  secret information to exchange a secret key
• An eavesdropper can passively snoop secret key as
  its being exchanged

8
Classical Cryptography

• Problems with Public-key encryption
• No key distribution problem
• However, security relies on unproven mathematical
  assumptions such as the difficulty of factoring
  large integers
• Shor has already shown that the assumption wont
  hold up against quantum computation

9
What can be done?

• Private key is vulnerable to classical attacks,
  Public key is vulnerable to quantum attacks
• Solution Augment private key encryption with
  quantum key distribution

10
A Slightly Different Problem

• Before worrying about eavesdroppers, lets
  consider at simple bit-commitment scenario
• Alice and Bob (who are mutually untrustworthy)
  wish to play a multiplayer game over a network
• Each player takes one action per round
• In order to prevent a player from waiting for the
  others move before deciding, each player commits
  themselves to an action by first transmitting an
  encrypted move (hash)
• Upon receiving the actual move, each player can
  encrypt it and compare to the previously received
  hash

11
Quantum Bit Commitment

• Why is bit commitment useful?
• Allows Alice to commit to a certain action
  without revealing that action to Bob
• Alice gives Bob a hint about what her action
  will be
• Later, if Alice wishes to reveal that action to
  Bob, the hint allows Bob to be certain that Alice
  has not changed the action

12
Quantum Bit Commitment

• Obvious problem with classical bit commitment
• Problem similar to the public-key encryption
  problem
• Hard for Alice to give Bob evidence that will
  both lock in her action AND prevent Bob for
  interpreting her action from the hint
• If we give Bob unlimited computational power (or
  a quantum computer) he could decrypt the hash and
  gain an advantage over Alice

13
Quantum Bit Commitment

• Solution
• We need a hash that is not based on a shared
  algorithm that Bob could reverse
• Add a quantum channel in parallel to the
  classical communication channel
• Utilize quantum channel for transmission of
  hint to Bob

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Quantum Bit Commitment

• Alice wants to commit a bit v to Bob
• Alice calls a function commit(v) that uses the
  quantum channel to transmit a hint to Bob
• Later, Alice calls unveil(v) to reveal v
• Bob can use his hint to ensure that Alice has not
  changed v

15
Quantum Bit Commitment

• Notation
• Rectilinear Base
• 0gt,1gt
• Diagonal Base
• (0gt1gt)/sqrt(2), (0gt-1gt)/sqrt(2) X
• Vectors
• 0gt
• 1gt -
• (0gt1gt) /
• (0gt- 1gt) \

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Quantum Bit Commitment

• Algorithm commit(v)
• Bob supplies Alice with a matrix G that generates
  code words that differ by at least 10en bits
• Alice chooses a random n-bit string(r)
• Alice uses matrix G to generate a codeword(c)
  such that rc v
• Alice announces r to Bob
• Alice chooses another n-bit random string(b)
• Alice sends c to Bob on the quantum channel
  encoded according to
• bi0 -gt 0 1 -
• bi1 -gt x 0 / 1 \

17
Quantum Bit Commitment

• Algorithm commit(v)
• Bob chooses his own random string of bases (b)
• Bob uses b to measure the incoming values which
  gives Bob string c
• Bob now has a quantum hint of what Alices
  codeword is
• However, Bob cannot get any information out of
  the codeword since he doesnt know what
  transmission bases Alice used
• Statistically Bob will only guess 50 of the
  bases correctly 50 of the codeword bits are
  effectively random

18
Quantum Bit Commitment

• Algorithm unveil(v)
• Alice sends c,b,v to Bob
• Bob calculates a compare-summation on code words
  c and c for all bits in which Bob correctly
  guessed the transmission basis.
• ? (bi bi) ? (ci xor ci) / (n/2)

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Quantum Bit Commitment

• Algorithm unveil(v)
• If rc v (Alices original setup)
• And c is a valid codeword of matrix G
• And ? lt 1.4e
• Bob accepts v
• Otherwise
• Bob rejects v

20
Quantum Bit Commitment

• Example
• Alice obtains a c 10110110
• Generates random base
• B XXXX
• Encodes c in base B
• - \ - / \ - /
• Transmits quantum string to Bob

21
Quantum Bit Commitment

• Example cont
• Bob receives encoded string
• Chooses own random base and measures the quantum
  transmission
• B XXXX
• Obtains result
• c 0 1 0 0
• random result

22
Quantum Bit Commitment

• Example cont
• Alice sends b, c to Bob
• Bob compares b to b
• b XXXX
• b XXXX
• Bob compares c and c for bits corresponding to
  matches in b an b
• c 10110110
• c 0100

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Can Either Player Cheat?

• Alice
• In order to fool Bob, would have to alter her
  codeword so that rcnew v
• However, she also has to ensure that cnew is a
  valid codeword of generating matrix G
• This means Alice will have to flip at least 10en
  bits to reach a new, valid codeword
• Also, to avoid detection, all Alices bit flips
  would have to be done on bits in which Bob chose
  a different measurement base than Alice did

24
Can Either Player Cheat?

• Alices Chances
• The Probability that a given base differs
• Prob (b ! b) 0.5
• Prob (success) (0.5)10en
• So for
• N 1000, e 1
• P(success) 7.9 10-31

25
Can Either Player Cheat?

• Favorable conditions for Alice
• If there is no noise on the channel when Alice
  transmits Bob can attribute some of the
  differences to noise
• Alice could afford to incorrectly flip X bits
  where x must be
• 0.7en gt X 7 bits in our previous example
• Improves her chances to 1 10-28
• Does not help when n is large enough
• Flipping bases in conjunction with bits can also
  help

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Can Either Player Cheat?

• Bob
• Until Alice reveals b, Bob knows nothing about c
  since c is nothing but random data until the
  bases are known.
• The information hidden in c comes into being
  only when Alice reveals her quantum transmission
  bases.
• Since no information exists prior to Alices
  transmission, its impossible for Bob to draw
  information out of c

27
Can Either Player Cheat?

• Bob does an exhaustive key search
• Restricting Alices codeword choices could help
  Bob
• Bob finds all possible code words from matrix G
  that differ by 0.25n bit flips from the measured
  codeword
• However code words themselves only differ by 10en
  which would produce an large enough set to negate
  Bobs efforts

28
Can Either Player Cheat?

• Bob uses a non-standard base
• Uses base halfway between diagonal and
  rectilinear
• Still only gives Bob 75 bit accuracy
• Also negates Bobs ability to check Alices moves

29
Conclusions

• With a significantly large n and a reasonable e,
  a cheat-proof Bit commitment algorithm can be
  implemented
• Using a Quantum channel allows a sender to
  create information after it has been
  transmitted
• Bobs random data contains no information until
  Alice announces her transmission bases
• Will be a useful property for Quantum Key
  Distribution
• The bothersome properties of quantum mechanics
  ensure that the algorithm works
• If measurement did not destroy quantum
  information, Bob could continue to measure the
  bits received until he was probabilistically
  certain of the correct value
• Likewise if Bob could clone quantum states