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PRIME FACTORIZATION AND ITS
APPLICATION TO CRYPTOGRAPHY
Ricardo Estrada
Justin Griffin
Vishaal Sharma
WHAT IS PRIME FACTORIZATION?
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It is the breaking down of a composite number into smaller non
trivial divisors, which when multiplied together equals the original
integer.
A composite number is any positive integer greater than one that
is not a prime number.
A Prime Number can be divide evenly only by 1 or itself and it
must be a whole number greater than 1.
List of some of the prime numbers.
HOW IS PRIME FACTORIZATION FOUND?
Example of Prime Factorization
There are different methods which can be utilized to find the prime
factorization of a number.
One way is to repeatedly divide by prime numbers
Prime factorization of 80?
Start working from the smallest prime number, which is going to be 2 in
this case.
80 / 2 = 40
80 is evenly divided by 2 but 40 is not a prime number so we continue
40 /2 = 20
40 is evenly divided by 2 but 20 is not a prime number so we continue
20 / 2=10
20 is evenly divided by 2 but 10 is not a prime number so we continue
10 / 2 = 5
10 is evenly divided by 2 and 5 is a prime number so we have the answer
2X2X2X2X5=80
ANOTHER METHOD
Another way to approach the task is to choose any pair of factors and split these
factors until all the factors are prime:
Break 80 into 8 X 10
The prime factors of 8 are 2, 2 and 2
The prime factors of 10 are 2 and 5
So the prime factors of 80 are 2, 2, 2, 2, and 5
WHAT IS CRYPTOGRAPHY?
The art of protecting information by transforming it (encrypting it) into an
unreadable format, called cipher text. Only those who possess a secret key
can decipher (or decrypt) the message into plain text.
Modern cryptography concerns itself with the following four objectives:
1) Confidentiality (the information cannot be understood by anyone except
for whom it was intended for)
2) Integrity (the information cannot be altered in storage or transit between
sender and intended receiver without the alteration being detected)
3) Non-repudiation (the creator/sender of the information cannot deny at a
later stage his or her intentions in the creation or transmission of the
information)
4) Authentication (the sender and receiver can confirm each other's identity
and the origin/destination of the information)
WHY IS PRIME FACTORIZATION USED IN
CRYPTOGRAPHY?
Cryptography is all about number
theory, and all integer numbers are
made up of primes, so you deal with
primes a lot in number theory. More
specifically, some important
cryptographic algorithms such as RSA
critically depend on the fact that prime
factorization of large numbers takes a
long time. Basically you have a "public
key" consisting of a product of two large
primes used to encrypt a message, and a
"secret key" consisting of the primes
used to decrypt the message. You can
make the public key public, and
everyone can use it to encrypt messages
to you, but only you know the prime
factors and can decrypt the messages.
Anyone else who wanted to access your
intended crypted messages would have
to factor the number, which takes too
long to be practical.
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What are its benefits?
Used for encryption of data allowing only the person with the key to access that data, ensuring
privacy.
Used for signing messages and proving to the recipient that you are who you say you are.
Used to solve mathematical problems.
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What are its cons?
Because governments do not wish certain entities in and out of their countries to have access to
ways to receive and send hidden information that may be a threat to national interests,
cryptography has been subject to various restrictions in many countries, ranging from
limitations of the usage and export of software to the public dissemination of mathematical
concepts that could be used to develop cryptosystems. However, the Internet has allowed the
spread of powerful programs and, more importantly, the underlying techniques of
cryptography, so that today many of the most advanced cryptosystems and ideas are now in the
public domain.
An issue that arises in the usage of prime factorization to cryptography is the potential abuse
by people with negative objectives. Often times, cryptography can be used by criminals to
conceal the data they are sending back and forth through the internet. These criminals range
from sexual predators who are trying to conceal any data that might get them in trouble, drug
dealers, terrorists or people that are committing crimes and are trying to conceal it from the
prying eyes of law enforcement.
This abuse of cryptography brings up the current concern about the government wanting the
keys to all encryption software. This means that they would have the ability to intercept
anybody's data, decrypt it and observe the message. This is clearly a violation of our privacy.
The current work around is to use freely available encryption software or use software that
comes from overseas, since those companies don't have to worry about following our
government regulations; and its easily available online.
WHEN DID IT FIRST GET USED?
The origin of cryptography is usually dated from about 2000 BC, with the Egyptian practice of
hieroglyphics. These consisted of complex pictograms, the full meaning of which was only known to an
elite few.
Looking further, the Spartans (around 5B.C.) developed a cryptographic device to send and receive secret
messages. The device was a small cylinder called a scytale. This scytale was in the possession of both the
sender and the recipient of the message. To prepare the message, a narrow strip of parchment or leather,
much like a modern-day paper streamer, was wound around the scytale and the message was written
across it. Once unwound, for transport to the receiver, the tape displayed only a sequence of meaningless
letters until it was re-wound onto a scytale of exactly the same diameter and the message could then be
read.
Julius Caesar(100 - 44BC)did not trust his messengers when communicating with his governors and
officers; For this reason, he created a system in which each character in his messages was replaced by a
character three positions ahead of it in the Roman alphabet.
Cryptography changed the way we fought wars and led us to enter WWI.
For example, The Zimmermann telegram:
On the first day of hostilities, the British cable ship TELCONIA located and cut Germany's transatlantic
cables, forcing them to send all their international traffic via Sweden or American-owned cables. Both
means ran through the UK and soon all German traffic was routinely routed to ROOM 40, the Royal
Navy's cypher organisation.
On or about January 16, 1917 two Room 40 cryptanalyst William Montgomery and Nigel De Gray , were
given a message encrypted in the German Foreign Office code, a book cypher number 0075. By the next
morning they had deduced enough of the message to be chilled by its content. Sent by the German Foreign
Minister Zimmermann to the Mexican President via the German Embassies in Washington and Mexico
City, it advised the President of Mexico that Germany proposed to start unrestricted submarine warfare in
February and that he should, with German help, attack the US and also convince the Japanese to do the
same.
In short order the full text was recovered and presented to US President Wilson. On April 2, 1917 the then
neutral US declared war on Germany and by 1918 Germany had been defeated.
This in turn leads to accelerated sociological and technological change. The first world war showed the
importance of cryptography on the battlefield, and the danger of weak encryption.
The second world war became a defining moment in the history of cryptography and placed it squarely at
the centre of military and political strategy from that time to the present day.
HISTORY OF PRIME NUMBERS IN CRYPTOGRAPHY
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Although not used in cryptography, prime
numbers first show up in recorded history in
ancient Greece.
The Sieve of Eratosthenes is developed by ancient
Greek mathematician Eratosthenes.
 This was around the year 300 BC!
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Prime numbers then disappear from history for
the next two thousand years, reappearing again
in the 17th century
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Pierre de Fermat developed Fermat’s little theorem.
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22n + 1 are prime numbers. (Unproved at the time).
French monk Marin Mersenne used the formula 2p-1,
with p being a prime number.
PRIME FACTORIALS IN THE MODERN AGE
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In modern computing, the first
cryptographic standard in the 1970s
was NES.
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Everything changed in 1976 when the
Diffie-Hellman key exchange was
introduced by Whitfield Diffie and
Martin Hellman.
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This standard did NOT use prime
numbers. Instead, it used a block cypher.
Block cyphers essentially have plaintext at
the root of their encryption.
This was a method for two parties with no
prior knowledge of each other to share
and establish a secret key.
This was radically new concept at the time
and went a long way into improving
cryptography.
The original implementation of this
involved the formula Ya=G^(Xa)(mod
P), where P is a prime number and G
is primative root mod P.
This was the beginning of public key
cryptography.
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Later, in 1978 cryptography using
primes comes into its own with the
RSA algorithm.
RSA is used in many different things,
from credit cards to browsers to email
encryption.
THE DIFFIE-HELLMAN KEY EXCHANGE
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Definition: If every element
in group G can be expressed
as the product of finitely
many powers of g, then
element g is a generator.
Definition: Now assume g is
a primitive root mod p. If
g^x ≡ y (mod p) then the
discrete logarithm of y to the
base g is indg(y)=x mod(ø(p))
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Two people, A & B, do not
want C to hear their message.
They create two public
numbers g and p, where p is a
prime and g is a primitive
root mod p.
A now computes u ≡g^a(mod
p) and sends u to B, where a
is another random number A
has chosen.
B does the same thing,
choosing random number b
and sending v from
v ≡g^b(mod p) to A.
Now both have the same key,
computed with k=g^(ab)(mod
p)
THE RSA ALGORITHM
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Created by Rivest, Shamir, and
Adleman at MIT in 1977.
Users A and B have the same
problem as in the previous slide, not
wanting C to have their information.
Both select two large prime numbers
at random, p and q of near equal size
so that n = pq is 1024 bits.
Now compute n=pq and φ= (p-1)(q-1).
Integer e is then chosen, such that 1
< e < φ, & gcd(e, φ) = 1.
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Now compute the secret exponent d
such that 1 < d < ≡ & ed ≡ 1 (mod φ).
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Note that the most common choices for
e are 3, 17 and 65537. All of these
numbers are Fermat primes
(Fx=2^(2^x)+1).
The Extended Euclidean Algorithm is
used to do this, and the process is
known as modular inversion.
The public key is (n, e) and the
private key is (n, d). d, p, q, and φ are
kept secret.
From the above, it is easy to see why
the RSA algorithm is fairly secure.
This is just the key generation!
A BETTER PICTURE OF RSA
Key Generation
Encryption
Decryption
FUTURE OF PRIME FACTORIZATION AND
CRYPTOGRAPHY
In theory, a newer technology could render current cryptographic schemes (such as
RSA) useless. Current cryptographics use the process of prime factorization to
select a number so large that it would be impossible for any one to break the cypher
text. With todays computing power, it would take millions of years before a
computer could decipher the text.
The next big threat to current cryptography schemes is the quantum computer,
which is currently being researched by various universities throughtout the world.
A large quantum computer, if one is ever built, could theoretically factor numbers
quickly enough to defeat the code, which would make this cryptographic and its
prime factorizatioin useles.
If a quantum computer is eventually built, the prime factorization algorithms being
used for encryption would be rendered useless. However, there will be alternate
cryptographic schemes that employ different algorithms that do not use prime
factorization. It seems that the future of prime factorization and its application to
cryptography may be coming to an end due to the processing capabilities of
quantum computing. If a normal computer would take billions of years to decipher
a text, a quatum computer could theoretically decipher this text in minutes! Below
are the current methods being used to encrypt our data and the possible future
replacements. Hint, they do not include prime factorization.
POSSIBLE REPLACEMENTS:
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Hash based signature scheme
A hash function is an algorithm that transforms text into a relatively short
string of bits, called the signature. Its security is based on being able to
produce a unique signature for any given input. Even inputs that are only
slightly different from one another should produce different hashes.
Error correcting codes
Such a scheme would introduce errors into a message to make it unreadable.
Only the intended receiver of the message would have the right code to correct
the error and make the document readable.
Multivariable public-key cryptosystem (MPKC)
To crack it, a machine must solve multivariable nonlinear equations. This type
of cryptography is extremely efficient and much faster to produce than other
schemes, such as RSA, which must use numbers hundreds of digits long to be
secure.
Lattice based system
The lattice is a set of points in a many-dimensioned space. Possibly useful for
both digital signatures and public-key cryptography, a lattice system would be
cracked by finding the shortest distance between a given point in space and the
lattice.
It has potential, but it still needs more research.
EUCLID OF ALEXANDRIA
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Euclid of Alexandria was an Egyptian
Greek Mathematician around 300BC.
His book about mathematical elements
which consist of a collection of definitions,
theorems and mathematical proofs is one of
the most important works in the history of
mathematics.
“Euclid Elements also include number
theory. It consider the connection between
prefect numbers and Mersenne prime, the
infinitude of prime numbers, Euclid’s
lemma on factorization, and the Euclidean
algorithm for finding the greatest common
divisor of two numbers.”
REFERENCES
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Mathisfun.com. 2011. Date of access May 2, 2011
<http://www.mathsisfun.com/prime-factorization.html>
Faculty.etsu.edu. Bob Gardner. 2009. Date of access May 2, 2011
<http://faculty.etsu.edu/gardnerr/Geometry-History/euclid.htm>
wikipedia.org. 2011. Date of access May 1, 2011
<http://en.wikipedia.org/wiki/Prime_factor>
Wikipedia.org. 2011. Date of access May 1, 2011
<http://en.wikipedia.org/wiki/Euclid>
Arachnoid.com. Paul Lutus. 2008. Date of access April 28, 2011.
<http://www.arachnoid.com/prime_numbers/index.html>
Stallings William. Network Security Essentials Application and
Standards. Upper Saddle River,NJ: Prentice Hall, 2007.