David Froot
 How
do we transmit information and data,
especially over the internet, in a way that is
secure and unreadable by anyone but the
sender and recipient?
 Encryption is simply the process of
transforming information, such as plain text
or numbers, using an algorithm. Only those
with the algorithm can decipher the
encrypted information, and even a relatively
simple encryption technique can be difficult
to decode without the key.

Original encryption methods were simple. For example a
note would be encrypted using a new alphabet. A letter
shift was a common method:
ABCDEFGHIJKLMNOPQRSTUVWXYZ
ABCDEFGHIJKLMNOPQRSTUVWXYZ


A is now represented by an E, B by an F, etc. “The Quick
Brown Fox” becomes “Xli Uymgo Fvsar Isb”
But:
 This simple cipher is easily cracked by powerful
computers
 This cipher requires both the sender and recipient
have the decipher – but you cannot transmit encrypted
data to someone who doesn’t already have the cipher
code.

If both sender and recipient have the hashing algorithm
key, called symmetric key encryption, then data can be
sent safely. Key encryptions are based on the idea of
“hash values” and “hashing algorithms”

It would be impossible to determine from the hash value
that the input was 10,667 and the algorithm was a
multiplication of 143. Even this incredibly simple
algorithm is hard to break.
Modern algorithms are more complex and produce 128 bit
hash values which have
3,402,823,669,209,384,634,633,746,074,300,000,000,000,0
00,000,000,000,000,000,000,000,000 combinations and

makes the hashing algorithm impossible to compute.
 What
is both the sender and recipient don’t
have the key?
 We need a way of encrypting and decrypting
without sharing a key.
 Today, the standard practice is called Public
Key Encryption and uses a pair of two keys.
One key is privately held by one person, and
the other key is publicly known. The public
key can encrypt data but not decrypt it.
Likewise the private key can decrypt data
but not encrypt it.

Two keys are generated. If Bob wants to sent
secure data to Alice, he encrypts using Alice’s
public key. Then only Alice can decrypt with her
private key.
Source:
http://en.wikipedia.org/wi
ki/Public-key_cryptography
But how are these keys generated? And what
mathematical algorithms allow you to encrypt but not
decrypt and vice versa? A simple multiplication
function like the above hashing algorithm clearly
doesn’t work. What does?

Source: http://www.freesoft.org/CIE/Topics/144.htm
We start with a new
number system
called a modulus.
This is simply a
number system that
repeats after
exceeding its
maximum value, in
this case 15, but
can be any number
at all. It forms a
“loop” rather than
a traditional
number “line”.
Arithmetic is
performed
relatively normally.
 Modulus’
have a few special properties which
are used to solve the encryption problem.
 The most important is called the Euler
Totient function, which says that for any
number x, and a special number q(m):
q(m)
x
 The
=x
value of q(m) is derived from the the
modulus limiting number, i.e. 15.
 When
the modulus limiting number m is the
product of two prime numbers p and q, then:

q(m) = (p-1)(q-1)+1
 q(m)
 So
can be rewritten as 2 factors: zy = q(m)
for any value x, we can raise it to a power
z: xz which becomes essentially a “random”
number. Only when (xz)y does it return the
original value x.
 If
x is any number we wish to encrypt, then
we can call z the public key, and y the
private key. A simple example:
 2 prime numbers 3 and 11: m = 33



Q(m) = (3-1)(11-1)+1 = 21 (factors 3 and 7)
If we want to encrypt the number 20 with public
key:
203 = 8000 = 14 in mod 33.
 Given
a value 14, we need the private key
(7) to decrypt this number. The public key
(3) does nothing to help decrypt 14.

147 = (203)7 = 105413504 = 20 in mod 33
 This
simple example demonstrates the
concept but is easy for a computer to
decode.
 For large values of m, solutions are nearly
impossible to compute because even though
the value of m is known, no efficient
algorithms exist to determine prime factors
of a very large number. Even if q(m) were
computed, the factors chosen for y and z are
arbitrary and difficult to compute.
 http://computer.howstuffworks.com/encrypt
ion2.htm
 http://en.wikipedia.org/wiki/Encryption
 http://www.freesoft.org/CIE/Topics/144.ht
m
 http://en.wikipedia.org/wiki/Publickey_cryptography
 http://en.wikipedia.org/wiki/Blowfish_(ciph
er)