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Quantum Algorithm
and its applications
Ritajit Majumdar
MTech 2nd YearRoll Number : 97/CSM/140001
Registration Number : 0029169 of 2008-2009
3rd semester seminar talk
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1 Introduction
2 Lets Go Quantum
3 Consequence of Shor Algorithm
4 So what Now????
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Introduction
1 Introduction
2 Lets Go Quantum
3 Consequence of Shor Algorithm
4 So what Now????
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Introduction Factorisation is hard
How it all began
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I d i F i i i h d
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Introduction Factorisation is hard
How it all began
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I t d ti F t i ti b d d t i d fi di
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Introduction Factorisation can be reduced to period finding
Factorisation to Period Finding
Let N = 15, x = 2
r 0 1 2 3 4 5 6 7 8 ...xr (mod N) 1 2 4 8 1 2 4 8 1 ...
The period of repetition is one less than a factor of N
The problem ofPrime Factorization can be reduced to the problem ofPeriod Finding
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 5 / 23
Introduction Factorisation can be reduced to period finding
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Introduction Factorisation can be reduced to period finding
Factorisation to Period Finding
Let N = 15, x = 2
r 0 1 2 3 4 5 6 7 8 ...xr (mod N) 1 2 4 8 1 2 4 8 1 ...
The period of repetition is one less than a factor of N
The problem ofPrime Factorization can be reduced to the problem ofPeriod Finding
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 5 / 23
Introduction Factorisation can be reduced to period finding
http://find/ -
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Introduction Factorisation can be reduced to period finding
Factorisation to Period Finding
Let N = 15, x = 2
r 0 1 2 3 4 5 6 7 8 ...xr (mod N) 1 2 4 8 1 2 4 8 1 ...
The period of repetition is one less than a factor of N
The problem ofPrime Factorization can be reduced to the problem ofPeriod Finding
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 5 / 23
Introduction Factorisation can be reduced to period finding
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Introduction Factorisation can be reduced to period finding
Is this reduction any good?
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 6 / 23
Lets Go Quantum
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Let s Go Quantum
1 Introduction
2 Lets Go Quantum
3 Consequence of Shor Algorithm
4 So what Now????
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 7 / 23
Lets Go Quantum Quantum Fourier Transform
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Q Q
Quantum Fourier Transform
FN= 1N
1 1 1 1 . . . 11 w w2 w3 . . . wN1
1 w2 w4 w6 . . . w2(N
1)
... ...
... ...
. . . ...
1 wN1 w2(N1) w3(N1) . . . w(N1)2
Hence FNij is wij
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 8 / 23
Lets Go Quantum Quantum Fourier Transform
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Q Q
Quantum Parallelism
Let N = 4 w=exp(i2 ) =cos(2 ) +i.sin(2 ) =i
Consider state|1 = 0 1 0 0T
F4 |1 = 12
1 1 1 11 i 1 i1 1 1 11
i
1 i
0100
= 12
1i
1
i
= 12 |0 + i2 |1 12 |2 i2 |3
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 9 / 23
Lets Go Quantum Quantum Fourier Transform
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Quantum Parallelism
Let N = 4
w=exp(i2 ) =cos(
2 ) +i.sin(
2 ) =i
Consider state|1 = 0 1 0 0T
F4 |1 = 12
1 1 1 1
1 i 1 i1 1 1 11 i 1 i
0
100
= 12
1
i1i
= 12|0
+ i2
|1
12
|2
i2
|3
Inference
Quantum Fourier Transform produces an equal superposition of all thebasis states
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Lets Go Quantum The Algorithm
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Shor Algorithm
Choose qsuch that N2
q
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Shor Algorithm
Choose qsuch that N2
q
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Shor Algorithm
Choose qsuch that N2
q
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Shor Algorithm
Choose qsuch that N2
q
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Shor Algorithm
Choose qsuch that N2
q
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Shor Algorithm
Choose qsuch that N2
q
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Shor Algorithm
The first register is in state - 1qr
qr1j=0|j.r+l
Apply QFT again 1r
r1k=0w
kl |k.qr
Measure the statecollapses to k.qr
where 0 k r 1
If gcd(k, qr) = 1computing gcd(k.q
r, q) gives q
r
q is already knownfind r
If gcd(k, qr)= 1, repeat the algorithm
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 11 / 23
Lets Go Quantum The Algorithm
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Shor Algorithm
The first register is in state - 1qr
qr
1
j=0|j.r+l
Apply QFT again 1r
r1k=0w
kl |k.qr
Measure the statecollapses to k.qr
where 0 k r 1
If gcd(k, qr) = 1computing gcd(k.q
r, q) gives q
r
q is already knownfind r
If gcd(k, qr)= 1, repeat the algorithm
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 11 / 23
Lets Go Quantum The Algorithm
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Shor Algorithm
The first register is in state - 1qr
qr
1
j=0|j.r+l
Apply QFT again 1r
r1k=0w
kl |k.qr
Measure the statecollapses to k.qr
where 0 k r 1
If gcd(k, qr) = 1computing gcd(k.q
r, q) gives q
r
q is already knownfind r
If gcd(k, qr)= 1, repeat the algorithm
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 11 / 23
Lets Go Quantum The Algorithm
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Shor Algorithm
The first register is in state - 1qr
qr
1
j=0|j.r+l
Apply QFT again 1r
r1k=0w
kl |k.qr
Measure the statecollapses to k.qr
where 0 k r 1
If gcd(k, qr) = 1computing gcd(k.q
r, q) gives q
r
q is already knownfind r
If gcd(k, qr)= 1, repeat the algorithm
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 11 / 23
Lets Go Quantum The Algorithm
http://find/http://goback/ -
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Shor Algorithm
The first register is in state - 1qr
qr
1
j=0|j.r+l
Apply QFT again 1r
r1k=0w
kl |k.qr
Measure the statecollapses to k.qr
where 0 k r 1
If gcd(k, qr) = 1computing gcd(k.q
r, q) gives q
r
q is already knownfind r
If gcd(k, qr)= 1, repeat the algorithm
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 11 / 23
Lets Go Quantum The Algorithm
http://find/http://goback/ -
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Shor Algorithm
The first register is in state - 1qr
qr
1
j=0|j.r+l
Apply QFT again 1r
r1k=0w
kl |k.qr
Measure the statecollapses to k.qr
where 0 k r 1
If gcd(k, qr) = 1computing gcd(k.q
r, q) gives q
r
q is already knownfind r
If gcd(k, qr)= 1, repeat the algorithm
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 11 / 23
Consequence of Shor Algorithm
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1 Introduction
2 Lets Go Quantum
3 Consequence of Shor Algorithm
4 So what Now????
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 12 / 23
Consequence of Shor Algorithm Public Key Cryptography is insecure
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RSA is insecure
Run time of Shor algorithm : O((log N)2(log log N)(log log log N))[1]
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 13 / 23
Consequence of Shor Algorithm Public Key Cryptography is insecure
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Other public key ciphers...
Shors Algorithm has been experimentally realized using NMR[2]
It has been realised using Ion Trap with only 5 qubits[5]
Counter Argument: SO WHAT? We can use cryptosystems likeElliptic Curve, Elgamal etc.
These algorithms rely on Discrete Logarithm Problem
Well, no... Shor himself has given a quantum algorithm that can solveDiscrete Logarithm problem in polynomial time[1]
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 14 / 23
Consequence of Shor Algorithm Public Key Cryptography is insecure
O
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Other public key ciphers...
Shors Algorithm has been experimentally realized using NMR[2]
It has been realised using Ion Trap with only 5 qubits[5]
Counter Argument: SO WHAT? We can use cryptosystems likeElliptic Curve, Elgamal etc.
These algorithms rely on Discrete Logarithm Problem
Well, no... Shor himself has given a quantum algorithm that can solveDiscrete Logarithm problem in polynomial time[1]
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 14 / 23
Consequence of Shor Algorithm Public Key Cryptography is insecure
O h bli k i h
http://find/ -
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Other public key ciphers...
Shors Algorithm has been experimentally realized using NMR[2]
It has been realised using Ion Trap with only 5 qubits[5]
Counter Argument: SO WHAT? We can use cryptosystems likeElliptic Curve, Elgamal etc.
These algorithms rely on Discrete Logarithm Problem
Well, no... Shor himself has given a quantum algorithm that can solveDiscrete Logarithm problem in polynomial time[1]
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 14 / 23
Consequence of Shor Algorithm Public Key Cryptography is insecure
O h bli k i h
http://find/ -
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Other public key ciphers...
Shors Algorithm has been experimentally realized using NMR[2]
It has been realised using Ion Trap with only 5 qubits[5]
Counter Argument: SO WHAT? We can use cryptosystems like
Elliptic Curve, Elgamal etc.
These algorithms rely on Discrete Logarithm Problem
Well, no... Shor himself has given a quantum algorithm that can solveDiscrete Logarithm problem in polynomial time[1]
Inference
Public key cryptography is INSECURE against quantum attacks
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 14 / 23
Consequence of Shor Algorithm Private Key Cryptography is insecure too
P i t k i h
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Private key ciphers...
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 15 / 23
Consequence of Shor Algorithm Private Key Cryptography is insecure too
P i t k i h
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Private key ciphers...
A recent paper published on February 18, 2016 claims that Symmetric KeyCiphers can be broken in (n) time using Simons Algorithm[4]
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 15 / 23
So what Now????
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1 Introduction
2 Lets Go Quantum
3 Consequence of Shor Algorithm
4 So what Now????
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 16 / 23
So what Now???? Quantum Cryptography
BB84 Protocol
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BB84 Protocol
Use two conjugate basis
|+
=
{,
}and
|=
{,
}to establish a
secret key between two parties at a distance[3]
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So what Now???? Quantum Cryptography
Other variants of Quantum Key Distributions
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Other variants of Quantum Key Distributions
E91 protocol [Ekert, PRL 1991]
Semi Quantum QKD [Boyer, Kenigsberg and Mor, PRL 2007]
Device Independent (DI) QKD
Idea by Mayers and Yao [FOCS, 1998]
Measurement Device Independent (MDI) QKD [Lo, Curty and Qi,PRL, 2012]
Side Channel Free (SCF) QKD [Braunstein and Pirandola, PRL, 2012]
Fully Device Independent (FDI) QKD [Vazirani and Vidick, PRL, 2014]
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 18 / 23
So what Now???? Rebirth of Classical Cryptography
Post Quantum Cryptography
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Post Quantum Cryptography
Lattice-based cryptography (e.g., NTRU)
Multivariate cryptography (e.g., Rainbow)
Hash-based cryptography (e.g., Lamport, Merkle)
Code-based cryptography (e.g., McEliece, Niederreiter)
Supersingular ECC
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 19 / 23
So what Now???? Rebirth of Classical Cryptography
Acknowledgement
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Acknowledgement
I would like to acknowledge the following whom I referred to for preparingthis talk -
1 Pedagogical Talk on Death and Re-birth of Classical Cryptographyin Quantum Era by Dr. Gautam Paul, ISI Kolkata, at ISCQI 2016.
2 Online course on Quantum Mechanics and Quantum Computationby Prof. Umesh Vazirani of University of California, Berkeley, from
www.coursera.org.
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So what Now???? Rebirth of Classical Cryptography
Reference
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Reference
[1] Peter W Shor. Algorithms for quantum computation: Discrete logarithms and
factoring. In Foundations of Computer Science, 1994 Proceedings., 35th AnnualSymposium on, pages 124 134. IEEE, 1994
[2] Vandersypen et al. Experimental realization of Shors quantum factoringalgorithm using nuclear magnetic resonance. Nature 414, 883-887 (20 December2001)
[3] Charles H Bennett. Quantum cryptography: Public key distribution and cointossing. In International Conference on Computer System and Signal Processing,IEEE, 1984, pages 175 179, 1984
[4] Kaplan et al. Breaking symmetric cryptosystems using quantum periodfinding. arXiv:1602.05973, 2016
[5] Monz et al. Realization of a scalable Shor algorithm. Science 04 Mar 2016:
Vol. 351, Issue 6277, pp. 1068-1070
Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 21 / 23
So what Now???? Rebirth of Classical Cryptography
Image Courtesy
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Image Courtesy
Boromir: generator-meme.com/meme/one-does-not-simply/
Shor: www-math.mit.edu/shor/
Batman: bgr.com/2015/08/07/batman-movies-dc-ben-affleck/
Boromir facepalm: generator-meme.com/meme/own/
Grumpy cat: www.dailymail.co.uk
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So what Now???? Rebirth of Classical Cryptography
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http://find/http://goback/