# AiS Challenge Team Interim

Team Number: 53

School Name: Moriarty High School

Area of Science: Cryptology

Project Title: Cryptoanalysis

```Introduction
Cryptology is the art of encoding and decoding messages and has
been important throughout history.  In ancient days people used
concealment ciphers.  The message was there but somehow hidden. Another
way of encrypting in the early days was transposition ciphers.  You would
write a message in columns and you would read it in a diagonal direction.
Another way of encrypting is substitution ciphers.  In substitution
ciphers, numbers and symbols replace the letters.  Today, cryptology has
become a very important topic. In the world there is a high security need
to encrypt and decode messages. Through this expanding field of study, a
project is formed.

Purpose
For our project we plan to create a program that will encrypt a message in
a new way. In our design we will attempt to incorporate the use of
fractals to encrypt the message.  Presently, we are starting small and
learning to encrypt messages using substitution - a method of encryption
long known and used.  After we have experience with encrypting and
knowledge about fractals, we will combine our research to make a
fractal-encoding program.  The project could be advanced further by
decrypting the newly formed message by the reverse process in which it was
encrypted.

Proposed Method of Solution
Although this project will be difficult it can be accomplished. We will
begin by studying encryption and messages that are encrypted. Then we will
need to set up a math model to encrypt messages in a new and, or, more
efficient way. Next, we will implement parallel processing to speed up
encryption time. Finally, the project can be extended by setting up a way
to decode encryptions such as the one created.

Project Significance
This project will be important in today's National Security Issues. There
would be a great advantage if our country were able to encode information
in ways impossible to decode while decoding secret information that we
need to know. We find cryptology to be very interesting.  As young
children, the element of making something only a select few can understand
has been very exciting. Now, as young adults, we are still fascinated with
this idea, and want to continue it through the encryption program we are
generating.  We want to create a code that will present a unique way of
encrypting messages.

Mentors
Our team attended the Supercomputing conference in Glorieta near the end
of October.  While there, we asked many scientists for help with refining
our project to a more doable one.  One expert instructed us that we should
focus our project on a new way to encrypt messages.  He suggested using
fractals as the new way of encryption.  We have followed his advice and
looked up information on fractals as well as general encryption.  The
Challenge also set us up with a mentor who reviewed our abstract.  He
contacted us and we are making a list of questions for him.  We would like
to know if he has any background with fractals to add to our research or
any better ideas for encryption.  He instructed us to focus our project
and we want to ask him for suggestions about a simple fractal/encryption
program we can start to focus with.   We know that mentors are an
essential part of a successful project. We are also getting help form a
mentor at Sandia National Lab who suggested that we should start with a
simple method of encryption such as a substitution cipher. Once the
substitution cipher worked with our fractal encryption we should then try
a more complex cipher in its place.

Math Model / Research
Substitution
Substitution cipher is very simple to understand.  A substitution cipher
is when you substitute the regular alphabet for a "cryptobet."  This
cryptobet is a regular alphabet where the letters are replaced with other
letters or numbers.  Although there are no set equations for substitution
ciphers, one equation we tried was as follows.  Since each letter has a
corresponding ASCII number, we added 25 to the ASCII number.  This would
make A Z and R l.  Another substitution method is to put a formula to
change the first letter.  For example, with the first letter being in the
variable x changing into the variable y one could use a formula such as y
= (12*x - 17/8)^2.  This formula would also find a number correlating to a
letter, but it would be more difficult to decrypt because the formula is
so complex.  There are many substitution ciphers. We are currently writing
a simple program that switches the alphabet with a cryptobet so that we
will know how to implement substitution into our program with fractals.
Fractals
To add to our project, we plan to use fractals as a new way to encrypt
messages.  We need to discuss this with our mentor after completing other,
simpler encryption methods.  The information we have presently is as
follows. Fractals are usually simple equations that, when reiterated
numerous times, create chaotic pictures that sometimes look like real
life.  One very simple fractal is the Sierpinski Triangle.  This triangle
can be formed through many different ways.  One is in the form of a game.
The idea is to mark three points in three different colors and then color
two sides of a dice each color. You have to start with a spot and then
roll the die.  Move halfway towards the color you rolled and mark a point
of that color.  Continue to roll moving halfway towards the color you roll
each time.  Rather than a mess of points, this game forms a triangle like
the one shown at left.  Other fractals are formed through diverse ways.
The Mandelbrot set iterates the equation (z^2+c) with z fixed at 0 and c
varying.  It is a set of all of the sets of a different fractal set, the
Julia set.  The Julia set is the function f(z)=z^2+c when z is a point and
c is a complex number of any value.  There are an infinite number of
equations and thus an infinite number of fractals.  The fractals are
created by plotting each point on a screen using the formula.  The points
that end up going out of the screen are one color, depending on how many
iterations it takes to leave the screen.  The points from which the
formula never leaves the screen are a different color.  This creates a
picture.  The twist is that each point has a real x coordinate and an
imaginary y coordinate.  This makes numbers needing great accuracy, which
is why computers are so convenient to use to make fractals.
From this information we plan to create a program to use fractals in
cryptology.  Some ideas we have today include substituting the ASCII
number for each letter as the variable in the equation (this is usually
"c").  We could also first use a substitution method and then from our
results we would replace the variable with our substitution.  We need to
ensure our formula is random enough it is difficult to crack.
Although there is the possibility we will use a different math concept to
encrypt, fractals are an interesting subject that would be fun to work
with.  The options are endless with fractals because of their basis in
math.
```

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