Download presentation

Presentation is loading. Please wait.

Published byLillian Parvin Modified over 2 years ago

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 : Y A (mod p) Bob’s function : Y B (mod p) Private key is : Y AB = Y BA (mod p) Eve cannot compute Y AB from p, Y, Y A, Y B One-way function: f(A)=Y A (mod p) –easy to compute. f –1 (Y A ) 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 g r = 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,g a x -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 = g k (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 (n) = (p-1)(q-1) Select e such that 1 < e < (n) and gcd( (n),e) = 1 (e is made public too) Calculate d such that d*e = 1 mod (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 2n 2 <= q <= 4n 2 Choose x at random such that gcd(x,n)=1 Calculate the discrete Fourier transform of a table of x a 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(x r/2 - 1,n) and gcd(x r/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 = {|+>, |->} a i is encoded in basis X/Z if bit b i 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 | ψ akbk > 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 !!

Similar presentations

OK

Dr.Saleem Al_Zoubi1 Cryptography and Network Security Third Edition by William Stallings Public Key Cryptography and RSA.

Dr.Saleem Al_Zoubi1 Cryptography and Network Security Third Edition by William Stallings Public Key Cryptography and RSA.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on sports day games Ppt on history of earth Ppt on itc company profile Ppt on javascript events tutorial Ppt on art of war summary Ppt on food of different states of india Ppt on organisation of data for class 11 Ppt on weapons of mass destruction 2016 Ppt on rc coupled amplifier definition Unlock ppt online free