Cyber Security and Post Quantum Cryptography By: Professor Lili Saghafi

© Prof. Lili Saghafi , All Rights Reserved
1
Quantum Computers
New Generation of Computers
Quantum Computing and its
impact on Global Cyber Security
QUANTUM COMPUTERS-READINESS PLAN
Quantum-resistant algorithms
Professor Lili Saghafi
National Science Foundation (NSF)
The Government-University-Industry Research
(GUIRR )
Aug. 2019
@Lili_PLS

2
Agenda
• Classical Computers Vs Quantum Computers
• Optimization and Database searches
• Superposition & Entanglements
• How we build a Quantum Computers
• Cyber security and Cryptography
• QUANTUM COMPUTERS-READINESS PLAN
• RSA
• RSA problem
• SSL/TLS encryption
• (RLWE) problem
• Conclusion 2

3
Introduction
• Traditional old-fashioned digital computers run on
data that is encoded according to the binary system.
• In binary, the state of any single bit can only be 0 or
1.
• The options are quite literally binary.
• Any single computing bit can only reside in one of
two positions.
• Now emerging as the next generation of computing,
quantum computers run on data that comes in the
shape of qubits or quantum bits.

4
Introduction
• Quantum goes beyond binary by virtue of a
qubit’s ability to reside in more than one of
two positions.
• A qubit can represent a quantum state made
up of two or more values simultaneously,
called a superposition.
• A qubit’s superposition can also be
differentiated depending upon the context in
which it is viewed, so in basic terms we get
more computing power in the same space.

5
Introduction
• But quantum states are fragile and quantum
errors are notoriously difficult to measure, so
we need to treat this new power with respect.
• How then could this new thrust of computing
strength give us new tiers of power to
analyze IT systems at a more granular level
for security vulnerabilities and protect us
through more complex layers of quantum
cryptography?

6

7
When Classical Computers
are not good enough
• Classical computers are really bad in solving
certain problems and yet they can solve small
versions of those types of problems but by the
time the problem gets big enough to be
interesting it run out of computing
horsepower.

8
1-Optimization Problems
• Optimization is to find the best solution to a
problem among many possible solutions
• Picture of a table , how many different ways
are there to configure ten people around a
table? The answer is 10 factorial.
• 10 seems so small but 10 factorial is 3.6
million
• There's 3.6 million ways to arrange 10 people
for dinner.

9
1-Optimization Problems
• Every time we add one person to dinner table
the number of possible configurations grows
exponentially
• We can solve small versions of this problem
on classical machines but we don't solve big
problems, big versions, of this problem very
well at all

10

11
1-Optimization Problems ,
second example
• Second example is chemistry this
is a picture of a nitrogenase
enzyme
• We care about this enzyme it's an
important catalyst for the creation
of ammonia which is an important
component of food fertilizer
pharmaceuticals and many other
things
• In this enzyme i've called out
three molecules iron sulfide
clusters in this enzyme of
different sizes the one on the left
it's four iron atoms and four
sulphur atoms

12
NITROGENASE ENZYME

13
second example
• Believe it or not this is the biggest of those iron
sulfide clusters we can simulate on the biggest
supercomputer that we have
• It's so small , this is the biggest molecule of these
three that we can actually simulate on a classical
machine, and the reason is because to actually
simulate what's going on in that molecule, I have to
account for every electron repulsion and every
attraction of the electrons of the nuclei and that
number grows exponentially , the bigger the
molecule ,every single electron exerts an
electrostatic force on every single other electron

14
second example
• When I add another one, I got to recalculate
all the electron energies ,so these two bigger
clusters they look so small we can't simulate
them
• There's actually many problems that have this
characteristic and what they have in common
is this idea of exponential scaling
• There's a short story about the power of an
exponential.

15
Grow Exponentially
• The creator of the game of chess brought the
chess board to the emperor and the emperor
said I love this game, what can I give you as a
reward
• The craftsman said okay there's 64 squares on
the chess board, on day one give me one
grain of rice and every day after that double
the number of grains of rice

16

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Grow Exponentially
• So the first day emperor gives him one grain of rice,
the next day eight, and after a week he had a
teaspoon full of rice, but after a month he had the
rice production of a small country, and after the full
64 squares it was mount everest of rice .
• It grows really fast
• The number 64 doesn't sound that big but two to the
64 is an enormous number
• So why do we think quantum computing is actually
going to allow us to solve some of these problems?

18
Quantum Effects
• One of the effects that makes quantum
computers more efficient in solving those
problems is SUPERPOSITION
• Classical information is basically a string of
zeros and ones
• Everything that classical computing has
enabled is boiled down to a sequence of zeros
and ones

19
Quantum Effects, 1-Superposition
• So quantum computing or quantum information has
this property that the state can exist in a
superposition of 0 and 1 at the same time
• So not just 0 not just one, but a superposition of 0
and 1
• We can have complex superpositions of 0 and 1 so
you start to be able to explore much richer set of
states
• If one cubic can be in a superposition of two states,
then two qubits can be in a superposition of four
states, and three qubits can be in a superposition of
eight states.

20

21

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Quantum Effects, 1-SUPERPOSITION
• The possibility space you can explore is much
more interesting and complex in quantum
information
• Next diagram is showing you a superposition
of five cubits
• You can be in a superposition of 32 states
• Superposition is the first quantum effect

23

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Quantum Effects ,2-ENTANGLEMENT
• The second effect is entanglement
• You get two qubits and entangling them together
• Measuring the first qubit can tell you something
about what will happen when I measure the second
qubit
• Entanglement is the second property that gives
quantum information a really unique difference
• Together this allows us to totally change how we run
algorithms

25

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How Quantum Effects Works?
• Take the optimization case if I'm going to consider
three point six million possible ways of configuring
ten people at a table classically I have to consider
each one individually and then I have to compare
them
• Here's how quantum computing is going to solve
that problem
• you take your qubits you go into a superposition of
all the possible states, all the possible configurations,
and then when you encode the problem into your
quantum computer you're applying a phase on each
of the states , the phase is that kind of access
towards the center of that sphere you saw in the
previous chart

27

28
How Quantum Effects Works?
• You code a phase on each of the states and you know when
waves are in phase the amplitudes add and the waves are out
of phase they cancel
• when you have noise cancelling headphones what you're
doing is you're creating noise that's exactly out of phase with
the noise you're trying to cancel
• So in quantum computing you're gonna go into a
superposition of all these states when you encode the
problem onto the machine you're applying a phase on each of
the coupon each of the states and then, you're using
interference and you amplify some answers and you cancel
other answers eventually arrive at the solution

29
How Quantum Effects
Works?
• it's obvious why the number of qubit matter
• if I have one qubit I can be in two states and the more qubits I
can have ,I can be in a superposition of two to the end States
• but another important factor is this error rate
• I have to be able to control what's going on, in the qubits
• if I have really high errors and then all of my operations don't
work out as I expect them to then that's not going to really
work so we're promoting a new metric called quantum
volume
• if you increase the number of qubits you can get to higher
computational power but not if you have really high error
rates
• so we have to both move towards lower error rates and
higher cubic count

30
How do you actually build
a quantum computer
• This is how it should work theoretically how
do you actually go and do this in real life
• So first of all you have to have qubits that
work in such a way that you can harness
quantum mechanics

31

32
How do you actually build a
quantum computer
• We build basically artificial atoms
• Atoms behave quantum mechanically, we build an artificial
atom and we make it out of a superconducting josephson
junction coupled to a microwave resonator.
• It actually looks like on the chip you have these squares that
are your qubits and these squiggly lines are your microwave
resonators and inside of the qubit is a superconducting
josephson junction
• We got to cool this thing down to point zero
• Based upon the definitions of the centigrade scale and the
experimental evidence that absolute zero is -273.15ºc
• Five kelvin, where zero is absolute zero
• Room temperature is 300 this is significantly colder than
outer space
Kelvin to Celsius formula
°C =
K - 273.15

33

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How do you actually build a
quantum computer
• we talked to the qubits with microwaves
• this is how we talk to the qubits we have
inside of a dilution refrigerator which you'll
see on the next chart
• we have all of these microwave cables that
allow you to actually go and probe the qubits
with microwaves
• this is what it looks like in the actual lab so
these giant white cylinders these are dilution
refrigerators

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Dilution refrigerator of IBM QC

38
Inside D Wave

39
IBM QC

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Dilution refrigerator

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Dilution refrigerator

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QUANTUM COMPUTING
SOLUTION are good for:
• Finding optimization solutions
• Searching databases
• Encrypt codes

43
Quantum
Computing
Affects Cyber
Security

44
1-Speed affects Cyber
Security
• Quantum computing is a game-changing technology
for cyber security due to the inherent speed boost it
offers to solve complex mathematical problems.
• In traditional/Classical computing, when compared
with quantum, is effectively “brute-forcing”
(systematically enumerating all possible candidates
for the solution and checking whether each
candidate satisfies the problem's statement)
mathematical problems, until it arrives at a solution,
thus the more complex the question, the slower the
answer arrives.

45
What then …
– Quantum computing has the potential to
transform cyber security.
– Some encryption algorithms are thought
to be unbreakable, except by brute-force
attacks.
– Although brute-force attacks may be hard for
classical computers, they would be easy for
quantum computers making them susceptible
to such attacks.
– All financial institutions, government agencies
healthcare information are in danger.

46
BRUTE FORCE ATTACK
• A brute force attack is a trial-and-error
method used to obtain information such as a
user password or personal identification
number (PIN).
• In a brute force attack, automated software is
used to generate a large number of
consecutive guesses as to the value of the
desired data.

47
Traditional cryptography
• Traditional cryptography relies on the fact
that factoring large prime numbers is
mathematically complex and hackers
attempting to brute-force, to get an answer
needs a long time.
• For quantum computers, this kind of
factorization is where they excel, potentially
reducing the time to solve problems from
billions of years to a matter of seconds.
• We can now use that power to build more
complex protection layers

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FACTORIZATION
• Factorization or factoring, consists of writing
a number or another mathematical object as
a product of several factorsproduct of several factors
• Usually smaller or simpler objects of the same
kind.

49
So….
• What we need to remember is that the
majority of attacks in today’s threat landscape
target the user in one way or another and
social engineering plays large, if not larger,
than technical expertise.
• As long as a human can be persuaded to part
with a secret, in inappropriate circumstances,
all the cryptography in the world will not help,
quantum or not

50
2-Security of QC affects
Cyber Security
• We should also look more generally at the
types of data transformation operations we
can perform in quantum computers to exploit
effects that are not present in the classical
world of IT.
• Effects such as superposition and
entanglement offer information-processing
benefits, many of which can be meaningfully
applied to cryptography, such as improved
random number generation

51
Random Numbers
• Computers can generate truly random
numbers by observing some outside data, like
mouse movements or fan noise, which is not
predictable, and creating data from it.
• This is known as entropy.
• Other times, they generate “pseudorandom”
numbers by using an algorithm so the results
appear random, even though they aren’t.
• They’re an attempt to achieve an
unpredictable, random result.

52
Responsibility of QC
affects Cyber Security
• The responsibility for safe use is by no means
guaranteed.
• Quantum computers running what is known as
shor’s algorithm pose some risks to current
cryptography.
• Some encryption algorithms are thought to be
unbreakable, except by brute-force attacks.
• Although brute-force attacks may be hard for
classical computers, they would be easy for quantum
computers making them susceptible to such attacks

53
SHOR'S
ALGORITHM
• Shor's algorithm is a quantum computer algorithm for integer
factorization.
• Informally, it solves the following problem: Given an integer,
find its prime factors.
• It was invented in 1994 by the American mathematician Peter
Shor.
• On a quantum computer, to factor an integer, Shor's
algorithm runs in polynomial time.
• The basic gist of Shor's algorithm is the process of period-
finding which is done by the Quantum Fourier Transform
(QFT).
• The QFT takes some function and figures out the period of the
function.
• For example, if for all , then the function repeats itself every
10 values, and we can say that it has a period of 10.

54
Safety of QC affects Cyber
Security
• Current quantum computers require near
absolute zero temperature to be isolated from
interference like radio waves and noise, so
qubits keep their quantum mechanical state.
• All these requirements make it difficult and
expensive for non-nation state actors ,a
machine which could effectively and
efficiently run Shor’s algorithm – the most
complex quantum algorithm known – could
enable us to factorize large prime numbers
and do things we cannot even imagine today.

55
Resistance of QC affects Cyber Security
Quantum Computing Will Really Revolutionize
Cryptography
• Such a great computing power, however, will
present a huge challenge for cryptography in
the future as cyber criminals will be able to
target organizations with highly complex
quantum attacks.
• security specialists are currently developing
quantum-resistant algorithms, but we are yet
to see how quantum computing will really
revolutionize cryptography in the future

56
CRYPTOGRAPHIC ALGO
RITHMS
• Post-quantum cryptography (sometimes
referred to as quantum-proof, quantum-safe
or quantum-resistant) refers to
cryptographic algorithms (usually public-key
algorithms) that are thought to be secure
against an attack by a quantum computer.

57
Quantum Computing Will
Really Revolutionize
Cryptography
• What is special about random numbers from
quantum computing,
• and why their early prototypes are being used by
Swiss banks and governments, is that they can be
used to create a ‘one time pad’.
• This is a special kind of encryption key, that is
essentially unbreakable.
• Interestingly, one time pads, were first used in
World War One and are made exceptionally secure
by being used only once, for a single message, so
code breaking techniques simply won’t work

58
ONE-TIME PAD (OTP)
• In cryptography, the one-time pad (OTP) is an
encryption technique that cannot be cracked,
• It requires the use of a one-time pre-shared
key the same size as, or longer than, the
message being sent.
• In this technique, a plaintext is paired with a
random secret key (also referred to as a one-
time pad).

59
ONE-TIME PAD (OTP)

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Quantum Computing Will
Really Revolutionize
Cryptography
• While it will change most of the encryption
algorithms commonly used on the internet, it
is not true that quantum will break all
encryption.
• “The encryption systems that are used to
secure data stored in database records and
archives, such as legal documents, use a
different technique which quantum
computing has been unable to break, so far”

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• Since instead of just computing in a linear binary
way, with the presence or absence of an electrical
charge being converted into "bits" of zeros or ones,
Quantum Computers can take the rich quantum
properties of subatomic particles and turn them into
"Qubits" that can be both zero and one at the same
time.
• Quantum Computers could potentially run
simulations and solve problems that are far too big
for today's computers.

63
Downside of Quantum Computing
• A Quantum Computer could also break public
encryption
• keys used today to keep data safe.

64

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QUANTUM-READINESS PLAN
• Quantum-readiness plan, providing advice
about where vulnerabilities might be in the
quantum-computer era, and strategies and
tools that could be implemented now to make
any transition into that era much easier.

66
QUANTUM-READINESS
PLAN Solutions
Asymmetric Cryptography
VS
Symmetric Cryptography

67
1-RSA
• RSA is one of the first practical public-key
cryptosystems and is widely used for secure data
transmission.
• The encryption key is public and differs from
the decryption key which is kept secret.
• In RSA, this asymmetry is based on the practical difficulty
of factoring the product of two large prime numbers,
the factoring problem.
• RSA is made of the initial letters of the surnames of Ron
Rivest, Adi Shamir and Leonard Adleman, who first
publicly described the algorithm in 1977.
• Clifford cocks, an English mathematician, had developed
an equivalent system in 1973, but it was
not declassified until 1997.

68
RSA (Rivest–Shamir–
Adleman)
• RSA (Rivest–Shamir–Adleman) is
an algorithm used by modern computers to
encrypt and decrypt messages.
• It is an Asymmetric Cryptographic Algorithm.
• Asymmetric means that there are two
different keys.
• This is also called Public Key Cryptography,
because one of the keys can be given to
anyone.

69
In an asymmetric key encryption scheme, anyone can encrypt
messages using the public key, but only the holder of the paired
private key can decrypt. Security depends on the secrecy of the
private key.

70
RSA problem
• RSA problem , a user of RSA creates and then
publishes a public key based on the two
large prime numbers, along with an auxiliary
value.
• The prime numbers must be kept secret.
• Anyone can use the public key to encrypt a
message, but with currently published
methods, if the public key is large enough,
only someone with knowledge of the prime
numbers can feasibly decode the message.

71
An unpredictable (typically large and random) number is used to
begin generation of an acceptable pair of keys suitable for use
by an asymmetric key algorithm.

72
In the Diffie–Hellman key exchange scheme, each party generates
a public/private key pair and distributes the public key. After
obtaining an authentic copy of each other's public keys, Alice
and Bob can compute a shared secret offline. The shared secret
can be used, for instance, as the key for a symmetric cipher.

73
RSA problem
• Breaking RSA encryption is known as the RSA
problem; whether it is as hard as the factoring
problem, it remains an open question.
• Quantum Computers are good for Data
encryption.
• Code are information in very large number
768 bite number ,
• RSA code broken in 2010, it can take 3 years
for Digital Computers.
• 1024 bite code number it takes 3000 years for
Digital Computers, and for Quantum
Computers in a minute.

74
In this example the message is only digitally signed and not
encrypted. 1) Alice signs a message with her private key. 2) Bob
can verify that Alice sent the message and that the message has
not been modified.

75

76
The type of solutions that QC can
provide
• It was once believed that Quantum Computers could
only solve problems that had underlying
mathematical structures, such as code breaking.
• New algorithms have emerged that could enable
Quantum Machines to solve problems in fields as
diverse as weather prediction, materials science
and artificial intelligence.
• The ability of Quantum Computers to process
massive amounts of data in a relatively short amount
of time makes them extremely interesting to the
scientific community.

77

78
RSA

79
2-SSL/TLS encryption
• Now because of security vulnerability to
Quantum Computers , websites that use the
widespread SSL/TLS encryption standard
currently tend to make use of the RSA
algorithm, which mathematician Peter Shor
showed in 1994 could be easily broken by a
quantum computer.
• Shor’s approach could also be used to crack
Elliptic Curve Cryptography, another primitive
increasingly used with SSL/TLS.

80
RSA Signature Using
Open SSL

81
SSL/TLS Encryption Vs Elliptic Curve
Cryptography
• SSL certificates most commonly use RSA keys and the
recommended size of these keys keeps increasing (e.g., from
1024 bit to 2048 bit a few years ago) to maintain sufficient
cryptographic strength.
• An alternative to RSA is ECC.
• ECC stands for Elliptic Curve Cryptography, and is an approach
to public key cryptography based on elliptic curves over finite
fields .
• Both key types share the same important property of being
asymmetric algorithms (one key for encrypting and one key
for decrypting).
• ECC can offer the same level of cryptographic strength at
much smaller key sizes - offering improved security with
reduced computational requirements.

82
Elliptic Curve Cryptography
How does ECC compare to RSA?
• The biggest
differentiator
between ECC and
RSA, is key size
compared to
cryptographic
strength.

83
Elliptic Curve Cryptography
How does ECC compare to RSA?
• ECC is able to provide the same cryptographic
strength as an RSA-based system with much
smaller key sizes.
• For example, a 256 bit ECC key is equivalent to
RSA 3072 bit keys (which are 50% longer than
the 2048 bit keys commonly used today).
• The latest, most secure symmetric algorithms
used by TLS (eg. AES) use at least 128 bit keys,
so it makes sense that the asymmetric keys
provide at least this level of security.

84
ELLIPTIC CURVE
CRYPTOGRAPHY
• ECC is a type of Public Key Cryptography
• There are many types of public
key cryptography, and Elliptic Curve
Cryptography is just one flavor. ... These keys
are used to encrypt and decrypt data so that
anyone in the world can look at the encrypted
data while it is being transmitted, and be
unable to read the message.

85
Transport Layer Security
(TLS)
• Transport Layer Security (TLS) is the successor
protocol to SSL.
• TLS is an improved version of SSL.
• It works in much the same way as the SSL,
using encryption to protect the transfer of data and
information
• Unlike public-key encryption, just one key is used in
both the encryption and decryption processes.
• The difference between Asymmetric Cryptography
and Symmetric Cryptography .
• Once data has been encrypted with an algorithm, it
will appear as a jumble of ciphertext.

86

87

88
(RLWE) problem
• The research focuses on building a protocol using
one of the primitives currently thought to be difficult
for quantum computers to solve, called the “ring
learning with errors” (RLWE) problem.
• Practical application of this , is by seeing how to
design a key exchange protocol that’s suitable for
use in SSL/TLS and then implementing and testing it.
• Rather than multiplying large prime numbers
together as in RSA encryption, or using points on a
curve as in Elliptic Curve Cryptography, here the
mathematical operation is based on multiplying
polynomials together, then adding some random
noise.

89
RING LEARNING WITH ERRORS”
(RLWE) PROBLEM.
• Ring learning with errors (RLWE) is a
computational problem which serves as the
foundation of new cryptographic algorithms
designed to protect against cryptanalysis by
quantum computers and also to provide the
basis for homomorphic encryption.
• The RLWE problem can be stated in two
different ways: a "search" version and a
"decision" version.
• Both begin with the same construction

90
RLWE, ring learning with errors” (RLWE) problem.
• The result makes it “much harder” to crack.
• RLWE hasn’t been studied intensively enough to prove
that it would be any more secure against quantum
computers than the techniques currently in use, but the
primitive seems to be one of the better bets currently
out there.
• If after years of Cryptanalytic Research (Cryptanalysis is
the study of ciphertext, ciphers and cryptosystems with the aim of
understanding how they work and finding and improving techniques for
defeating or weakening them) no one manages to break it, then
it may achieve the corresponding levels of confidence
that the research community has in the difficulty of
currently accepted problems, like factoring or elliptic
curve discrete log.

91

92

93

94

95
Conclusion
• Quantum Computers change cyber security and
Cryptanalysis.
• A major advantage that RLWE (ring learning with
errors” (RLWE) problem) based cryptography has
over the original Learning With Errors (LWE) based
cryptography is found in the size of the public and
private keys.
• RLWE is “much harder” to crack than any other
algorithm
• More studies require .

96
References, Images Credit
• SAP market place https://websmp102.Sap-ag.De/home#wrapper
• Wikipedia , IBM
• Forbeshttp://www.Forbes.Com/sites/sap/2013/10/28/how-fashion-
retailer-burberry-keeps-customers-coming-back-for-more/
• Youtube
• Professor saghafi’s blog
https://sites.Google.Com/site/professorlilisaghafi/
• TED talks
• Tedxtalks
• Http://www.Slideshare.Net/lsaghafi/
• Timo elliot
• Https://sites.Google.Com/site/psuircb/
• Http://fortune.Com/
• Theoretical physicists john preskill and spiros michalakis
• Institute for quantum computing https://uwaterloo.Ca/institute-for-
quantum-computing/
• Quantum physics realisation data-burger, scientific advisor: J. Bobroff,
with the support of : univ. Paris sud, SFP, triangle de la physique, PALM,
sciences à l'ecole, ICAM-I2CAM 96

97
Max Planck Institute for Physics (MPP) http://www.mpg.de/institutes
D-Wave Systems
Frank Wilczek. Physics in 100 Years. MIT-CTP-4654, URL = http://t.co/ezfHZdriUp
William Benzon and David G. Hays. Computational Linguistics and the
Humanist. Computers and the Humanities 10: 265 – 274, 1976. URL
=https://www.academia.edu/1334653/Computational_Linguistics_and_the_Hum
anist
Stanislaw Ulam. Tribute to John von Neumann, 1903-1957. Bulletin of the American
Mathematical Society. Vol64, No. 3, May 1958, pp. 1-49, URL
= https://docs.google.com/file/d/0B-5-JeCa2Z7hbWcxTGsyU09HSTg/edit?pli=1
I have already discussed this sense of singualirty in a post on 3 Quarks Daily:
Redefining the Coming Singularity – It’s not what you think, URL
= http://www.3quarksdaily.com/3quarksdaily/2014/10/evolving-to-the-future-
the-web-of-culture.html
David Hays and I discuss this in a paper where we set forth a number of such far-
reaching singularities in cultural evolution: William Benzon and David G. Hays.
The Evolution of Cognition. Journal of Social and Biological Structures 13(4): 297-
320, 1990, URL
= https://www.academia.edu/243486/The_Evolution_of_Cognition

98
• Peikert, Chris (2014). "Lattice Cryptography for the
Internet". In Mosca, Michele (ed.). Post-Quantum
Cryptography. Lecture Notes in Computer Science. 8772.
Springer International Publishing. pp. 197–
219. CiteSeerX 10.1.1.800.4743. doi:10.1007/978-3-319-
11659-4_12. ISBN 978-3-319-11658-7.
• Shor, Peter (20 November 1994). Algorithms for quantum
computation: discrete logarithms and factoring. 35th
Annual Symposium on Foundations of Computer Science
• NIST https://csrc.nist.gov/Projects/Post-Quantum-
Cryptography/Post-Quantum-Cryptography-
Standardization
• Summer School on Real World Crypto and Privacy (Sibenik,
Croatia )

99
Thank you!
Great Audience
proflilisaghafi@gmail.com
https://professorlilisaghafiquantumcomputing.wordpress.co
m/

100
Quantum Computers
New Generation of Computers
Quantum Computing and its
impact on Global Cyber security
Quantum-resistant algorithms
National Science Foundation (NSF)
The Government-University-Industry Research
(GUIRR )
Aug. 2019
@Lili_PLS

Cyber Security and Post Quantum Cryptography By: Professor Lili Saghafi

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Cyber Security and Post Quantum Cryptography By: Professor Lili Saghafi