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Code-Breaking with a Quantum Computer Credit for ideas and examples: Prof. N. D. Mermin’s class Phys 681 / Comp Sci 483 “Quantum Computation” (A good class)

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...plus possibly many more outputs with other probabilities

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Weirdness of Quantum Mechanics Recall: Schrodinger’s cat is alive and dead simultaneously (before you “measure” – i.e. look inside the box) – state of being of the cat is a superposition of alive and dead |state of cat> = a | alive > + b | dead > Make a “measurement”: i.e. look inside box – find cat alive with probability |a| 2 and dead with probability |b| 2

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Quantum Computing “Qubits”: superposition of classical bits – like being in the state “0” and “1” simultaneously |state of Q computer > = a’ |0> + b’ |1> Measure the QC and measure 0 with probability |a’| 2 and 1 with probability |b’| 2 All of QC built up from gates that can change internal state to different superpositions (i.e. change a’ and b’ to different coefficients a’’ and b’’)

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RSA Encryption

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CECIL - chooses two primes, p and q -chooses a public exponent e: no factors in common with N = (p-1)(q-1) ALFBIJOU

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CECIL - chooses two primes, p and q -chooses a public exponent e: no factors in common with N = (p-1)(q-1) -sends M = pq and e along a public channel ALFBIJOU -chooses (plaintext) message x to be encoded -encodes according to y = x e (mod M)

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CECIL - chooses two primes, p and q -chooses a public exponent e: no factors in common with N = (p-1)(q-1) -sends M = pq and e along a public channel -meanwhile, computes decoder ed = 1 (mod N) ALFBIJOU -chooses (plaintext) message x to be encoded -encodes according to y = x e (mod M)

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CECIL - chooses two primes, p and q -chooses a public exponent e: no factors in common with N = (p-1)(q-1) -sends M = pq and e along a public channel -meanwhile, computes decoder ed = 1 (mod N) -decodes: x = y d (mod M) ALF BIJOU -chooses (plaintext) message x to be encoded -encodes according to y = x e (mod M)

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CECIL - chooses two primes, p and q -chooses a public exponent e: no factors in common with N = (p-1)(q-1) -sends M = pq and e along a public channel -meanwhile, computes decoder ed = 1 (mod N) -decodes: x = y d (mod M) ALF BIJOU -chooses (plaintext) message x to be encoded -encodes according to y = x e (mod M) Hopelessness of factoring M -> cannot hope to guess N or d.

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CECIL - chooses two primes, p and q -chooses a public exponent e: no factors in common with N = (p-1)(q-1) -sends M = pq and e along a public channel -meanwhile, computes decoder ed = 1 (mod N) -decodes: x = y d (mod M) ALF BIJOU -chooses (plaintext) message x to be encoded -encodes according to y = x e (mod M) Hopelessness of factoring M -> cannot hope to guess N or d. Quantum computer finds the period r of y r = 1 (mod M) (i.e. lowest r for which this is true)

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CECIL - chooses two primes, p and q -chooses a public exponent e: no factors in common with N = (p-1)(q-1) -sends M = pq and e along a public channel -meanwhile, computes decoder ed = 1 (mod N) -decodes: x = y d (mod M) ALF BIJOU -chooses (plaintext) message x to be encoded -encodes according to y = x e (mod M) Hopelessness of factoring M -> cannot hope to guess N or d. Quantum computer finds the period r of y r = 1 (mod M) (i.e. lowest r for which this is true) Then calculate alternate decoder d’ via ed’ = 1 (mod r) and then can decode: x = y d’ (mod M)

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How Period-Finding Can Break RSA Encryption – A Quantum Algorithm

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Quantum (Shor’s) Algorithm each coefficient depends on y^r (mod pq) n = number of bits used in the computer j = some integer r = period (order)

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