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Breaking RSA Encryption with Quantum Computers: Understanding Shor's Algorithm

This overview delves into the intriguing world of quantum computing and its potential to disrupt traditional encryption methods, particularly RSA. It explores the fundamental concepts of superposition and entanglement, referring to Schrodinger's cat as a metaphor for quantum states. The piece outlines how quantum computers use qubits, enabling parallelism in computations. It further explains Shor's algorithm, which efficiently factors large numbers, thereby jeopardizing the security of RSA encryption. Ultimately, this discussion emphasizes the transformative capabilities of quantum technology on cryptography.

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Breaking RSA Encryption with Quantum Computers: Understanding Shor's Algorithm

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  1. 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)

  2. ...plus possibly many more outputs with other probabilities

  3. 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

  4. 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’’)

  5. RSA Encryption

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

  7. 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 ALF BIJOU -chooses (plaintext) message x to be encoded -encodes according to y = xe (mod M)

  8. 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) ALF BIJOU -chooses (plaintext) message x to be encoded -encodes according to y = xe (mod M)

  9. 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 = yd (mod M) ALF BIJOU -chooses (plaintext) message x to be encoded -encodes according to y = xe (mod M)

  10. 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 = yd (mod M) ALF BIJOU -chooses (plaintext) message x to be encoded -encodes according to y = xe (mod M) Hopelessness of factoring M -> cannot hope to guess N or d.

  11. 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 = yd (mod M) ALF BIJOU -chooses (plaintext) message x to be encoded -encodes according to y = xe (mod M) Hopelessness of factoring M -> cannot hope to guess N or d. Quantum computer finds the period r of yr = 1 (mod M) (i.e. lowest r for which this is true)

  12. 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 = yd (mod M) ALF BIJOU -chooses (plaintext) message x to be encoded -encodes according to y = xe (mod M) Hopelessness of factoring M -> cannot hope to guess N or d. Quantum computer finds the period r of yr = 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 = yd’ (mod M)

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

  14. 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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