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Quantum Cryptography ( EECS 598 Presentation)

Quantum Cryptography ( EECS 598 Presentation). by Amit Marathe. Outline. Classical Cryptography Private vs. Public Key Cryptosystem Classical Key Distribution Quantum Code-breaking Quantum Key Distribution. References.

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Quantum Cryptography ( EECS 598 Presentation)

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  1. Quantum Cryptography(EECS 598 Presentation) by Amit Marathe

  2. Outline • Classical Cryptography • Private vs. Public Key Cryptosystem • Classical Key Distribution • Quantum Code-breaking • Quantum Key Distribution

  3. References • P. Shor, “Algorithms for Quantum Computation: Discrete Logarithms and Factoring ”, Proceedings, 35th Annual Symposium on Foundations of Computer Science pp. 124-134. November 1994. • Nielsen and Chuang, “Quantum Computation and Quantum Information” • William Stallings, “Cryptography and Network Security: Principles and Practice”

  4. Classical Cryptography • Private Key Cryptosystem (Symmetric) - Secret key (same for encrypt/decrypt) - Encrypt/Decrypt algo may or may not be known - Examples: DES, AES, IDEA • Public Key Cryptosystem (Asymmetric) - proposed by Diffie, Helman [1976] - Encrypt/Decrypt Algo and Public key known - Examples: RSA, RC5

  5. Private vs. Public Key Algorithms • Public Key - Main disadvantage is that it is expensive in terms of computational power • Private Key - Faster and cheaper then Public Key - main disadvantage is that somehow we need to distribute the unique private key • Remember: Security depends on unproven mathematical assumptions -difficulty in factoring,finding discrete log etc.

  6. Classical Key Distribution • Use public key algorithm to distribute the private key • Example: Algorithms proposed by Diffie/Helman or Rivest et.al. (RSA) can be used to distribute the private key. How ?

  7. Classical Key Distribution (Diffie/Helman) • Alice and Bob choose Y and modulus p • Alice’s function : YA (mod p) • Bob’s function : YB (mod p) • Private key is : YAB = YBA (mod p) • Eve cannot compute YAB from p, Y, YA, YB • One-way function: f(A)=YA(mod p) –easy to compute. f –1 (YA) is called the “discrete logarithm” and is hard to compute

  8. Shor’s Discrete Log Algorithm Using Quantum Computation • Given prime number p , generator g of the multiplicative group (mod p) and x, we need to find r such that gr = x (mod p) • Choose a and b and create a superposition • Apply Fourier Transform to the above state to send a => c and b => d p-2 p-2 S = 1/(p-1) Σ Σ |a,b,gax-b(mod p)> a=0 b=0

  9. Shor’s Discrete Log Algorithm Using Quantum Computation • Probability of observing a state |c,d,y> with y = gk (mod p) is given by • Recover r from a pair c,d such that | 1/{(p-1)q} Σ exp {(ac+bd)2пi/q) |2 a,b,a-rb=k (mod p) -1/2q <= d/q + (r/q)(c – {c(p-1)}q /(p-1)) <= 1/2q (mod1)

  10. Classical Key Distribution (RSA) • Choose two prime numbers p and q (secret) • Calculate n = p*q (available to public) • Calculate f(n) = (p-1)(q-1) • Select e such that 1 < e < f(n) and gcd(f(n),e) = 1 (e is made public too) • Calculate d such that d*e = 1 mod f(n) • Public key KU = {e,n} • Private key KR = {d,n}

  11. Shor’s Factoring Algorithm Using Quantum Computing • Choose a smooth q such that 2n2 <= q <= 4n2 • Choose x at random such that gcd(x,n)=1 • Calculate the discrete Fourier transform of a table of xa mod n, order log(q) times

  12. Shor’s Factoring Algorithm Using Quantum Computing • Use a continued fraction technique to guess r • Two factors of n are then gcd(xr/2 - 1,n) and gcd(xr/2 + 1,n) • If the factors are 1 and n, try again.

  13. Quantum Key Distribution (QKD) • Protocol to create private key bits between two pairs over a public channel • Provably secure (conditioned only on fundamental laws of physics being correct) • Information gain implies disturbance - Eve cannot gain any information from the qubits transmitted from A to B without disturbing their state

  14. BB84 QKD Protocol • Alice creates two strings a and b of lengths (4+δ)n each • Basis X = {|0>, |1>} , Z = {|+> , |->} • ai isencoded in basis X/Z if bit bi is 0/1 • |ψ> = • Bob receives |ψ> from Alice • Alice and Bob discard those bits where Bob and Alice’s measurements differed -if less then 2n bits left then abort the protocol | ψakbk > k goes from 1 to (4+ δ)n

  15. BB84 QKD Protocol • Alice selects selects a subset of n bits (as the check bits) and conveys to Bob • Alice and Bob compare these n check bits. -If more then an acceptable number of bits disagree, protocol is aborted • Alice and Bob perform information reconciliation and privacy amplification on remaining n bits to obtain m private key bits

  16. Conclusions • Classical key distribution by using Public Key algorithms can be broken by Quantum Computing Algorithms • Quantum Key Distribution is provably secure ! (at least if fundamental laws of physics continue to hold) • Promising future for Quantum Cryptography !!

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