Below is a script which is in development for this series. ( UNDER CONSTRUCTION )

We begin with a problem.

Imagine that a girl named Alice has a treasure, and she needs to send a secret letter giving its location her best friend Bob. Alice is in hiding, so she asks her friend with a bike to deliver the message. Although she trusts her friend, there is always a chance that someone could steal it en route. To protect the treasure she must disguise the message and make it unreadable to anyone other than Bob.

Alice’s situation has reoccurred for thousands of years. It has decided wars and shaped our history. How did we solve it, say, 2000 years ago, and would we solve it any differently today?

The story of this problem begins with two ancient ideas, numbers and chance, inspired by two reoccurring needs, counting and gambling.

So we must return to prehistoric times, before Christ, the Greeks and civilization itself. 37,000 years ago: The stone age.

Chapter 1. From Counting Sticks to Primes: The nature of numbers

This chapter begins with two very simple problems, inspiring two very deep ideas.

Now to remember what is was like to during in Paleolithic times, we must think about what day to day life was like then. This is a time when our agenda was filled with one thing, survival. So problems related to survival is where we spent our mental and physical energy. And to survive, is to eat, so it is with food by which we can pin down a beginning to this story. Imagine two hunters. One is preparing a fire and the other is responsible for the catch of the day.

Eating in groups requires us to intuitively agree, or disagree, on what makes a fair share. In this case it is sharing a catch between two people. So we needed some concept of equal sharing, or piles. This is the essence of division. We can extend this idea of sharing to larger groups. If we have to share with more people we simple split our catch into smaller sized piles. To be unfair is to offer one person a larger pile at someones expense. This idea of fairness vs unfairness when sharing is the root of the modern concept of divisibility.

Hunting requires and understanding of both the short and long term patterns of animal movements. Imagine our hunters at night, trying to plan in advance for future attacks…how could they do this?  They needed some method of keeping track of time. This problem leads to the oldest human technology, that of a clock. The essence of a clock is a repeating pattern which we use to orient ourselves within the flow of time.

To find these repeating patterns our hunters looked up towards the heavens – filled with mysterious patterns.  The most obvious cycle we all recognize is the rising and falling of the sun to mark the passage of one day. However, this is a fairly short cycle. For counting long periods we looked to the moon.

The most beautiful & mysterious feature of the moon is the dynamic nature of it’s shape – it seems to grow and shrink over time. However there is a pattern one notices after prolonged observation,  every so often we experience the beauty of a full moon. Now this pattern repeats over a much longer, yet unchanging, period of time. Mathematics deals with the discovery, analysis and prediction of patterns – and to do “math” is to make abstractions. An abstraction occurs when we make a some observation of our physical world and simplify it using symbols. In this case we used a simple notch to represent an entire day. This is our earliest clock. A simple, yet powerful first step in our story.

This method of time keeping allowed the hunter to calculate that exactly 29 days pass between each full moon (1 month). However, when planning for future events we needed to figure out how many days were in half a month – this lead to a problem. Unfairness. If we divide 29 in half we end up with two unequal piles 14 and 15. Or, if we try to divide it into 4 piles we end up with the same problem: 7 7 7 8.

The only way to make equal piles of 29, is to have one pile of 29, or 29 piles of one. 29, in a sense, is unbreakable. Now we may ask more interesting questions such as how many numbers have this unbreakable property? How large can they be?

Well lets think like a philosopher. First we realize that we can divide all numbers into two distinct sets. The first contains all numbers which can be shared equally such as 6 or 9. This set of numbers is known as composite numbers. The other set contained those strange unbreakable numbers that could not be divided equally. Such as 7, 11 or 29. This set of numbers is known as prime numbers. By definition a prime numbers is divisible only by itself and 1.

The creation of these two categories is quite a striking idea.  If we list the entire universe of numbers, coloring the primes blue, we will find that they dance back and fourth between prime and composite in a strangely beautiful pattern.

There is no such thing as a biggest prime or composite, both go on forever. Here is a wonderful example of mathematical beauty, and what is most striking about this pattern is that it is never repeats…

Fast forward a few thousand years to Ancient Greece, where Euclid of Alexandria made some powerful discoveries about properties of numbers. He had an idea that all numbers, big and small, must share something in common. He continued where our hunters left off, equal piles and prime numbers. He thought about the universe of all whole numbers as comprised of primes which were unbreakable and non primes which were not. First he ignored all the primes, which we colored blue and focused on the leftover numbers which were not prime, the breakable numbers.

He noticed something interesting about this set of breakable numbers, known as composite numbers. Pick any number and break it down – imagine exploding the number apart into the smallest equal pieces. These are called factors and they are the building blocks for numbers. You can describe the number by adding these back together. However, Euclid noticed that these building blocks were always prime no matter what number we broke down – ah, prime factors.

Break any number into equal piles, you will always end up with prime factors.

15 = 3 + 3 + 3 + 3 + 3

25 = 5 + 5 + 5 + 5 + 5

49 = 7 + 7 + 7 + 7 + 7 + 7 + 7

However with large numbers as the list can get quite long.

254 = 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 ….

So Euclid imagined a way to multiply these prime factors instead of adding them to save space. Could he build any possible number by multiplying only primes together? To do this he first needed to find all the factors, not just the smallest. For example, take a number such as 30. Then breaking it down into all possible equal pieces.

30 = 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 +  2 + 2 + 2 + 2 + 2 + 2 + 2

30 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3

30 = 5 + 5 + 5 + 5 + 5 + 5

This gives us a list of all the building blocks known as prime factors. The prime factors of the 30, are 2, 3 and 5. The important point is that to build a number one needs to multiply all of its prime factors together in a special combination known as the prime factorization. In this case 30 has a prime factorization as:

30 = 2 * 3 * 5

Or for 15:

15 = 3 + 3 + 3 + 3 + 3

15 = 5 + 5 + 5

So we must multiply 3 and 5 together to get the valid combination:

15 = 3 * 5

Speaking of combinations, there is one big reason why this is important. Ever lock has one key or combination to open it, the same is true with numbers, ever number has exactly one prime factorization. So the prime factorization can be thought of as a secret key for a number. No two numbers will ever share the same key. This was one of Euclid’s great gifts to mathematics, this property was so important it still is being explored thousands of years later.  To make sure we understand this we should use a thought experiment.

Imagine that every number is a lock and every lock has a unique combination. The combination to each lock would be it’s prime factorization. The key to break open a number into it’s building blocks, is the prime factorization – The big question we now take away will return thousands of years later in our story….It is this, Given a massive number, how do quickly we find out the key?

348573945734895739573975985798435 = ? * ? * ? * ?

Chapter 2. Gambling: The secularization of chance

The arena, the roulette table, the tennis court, the race track… All are temporary worlds in which strategies and potential payoffs apply. A common element all of these worlds share is the element of chance. The word chance is derived from the Latin cadere meaning to fall. When something falls, such as dice, we seem to loose hope predicting its short-term fate. Through history it was common to assume that unpredictable events were controlled by divine intervention – the will of god.

However it wasn’t until 500 years ago that anybody even attempted to develop a theory of how to understand unpredictable (or random) events – known as probability theory. Probability theory was born out of a logical reflection on gambling.

In the 1550′s, an Italian mathematician named Gerolamo Cardano was known for his life long addiction to gambling. He was the first to figure out a way to calculate the probability of any random event in order to help him place the best possible wager. So, if the game he was playing required him to roll a pair he could figure out the exact odds of this event, offering him a huge advantage. Remember, at the time his opponent would have no clue such a thing was possible, instead assuming every roll was simply based on luck rather than logic. First, let’s pause for a moment and think about two important aspects of the physics of a dice roll.

First, balance. Dice must be constructed to be perfectly symmetrical, with a point of balance (known as center of gravity) in the middle of the dice.  An object which is not perfectly symmetrical will have a point of balance which is off center – such as an egg. If we roll an egg, it will always be drawn down towards it’s wide end and fall over on it’s broad side. A fair dice on the other hand will never favor a side.

Second, chaos. Why is the outcome of a dice roll uncertain or random?

Remember that it is not the dice itself which is unpredictable, it is it’s interaction with the environment. The speed, angle of the throw, the position of the tossing hand, the temperature, airflow, imperfections in the surface – a symphony of variables come into play every time we roll a dice. We can never roll it the exact same way twice, there are always tiny differences in the initial conditions of the toss. Tiny changes to the initial conditions are amplified as the dice bounces along it’s path becoming major changes, resulting in an outcome which is well mixed.

Predicting the outcome of dice is similar to predicting the position of a single molecule of cream after it drops into a hot cup of coffee. Once it hits the coffee it is bombarded by a symphony of collisions with the hot coffee molecules which mix the cream in all directions equally. After a few seconds millions of tiny collisions build on each other and mix the cream molecules around the entire volume of the coffee evenly. So after a certain period of time the original molecule is equally likely to be anywhere in the coffee – attempting to predict where it will be is simply out of our hands….

Now we can return to Cardano. He did something very interesting – he turned this idea of “equal likeliness” into a powerful recipe for calculating odds. He figured that to calculate the odds of any event, he must consider all possible outcomes. One dice roll has six possible outcomes, this is known as it’s probability space.

A powerful conceptual tool. Since every outcome is equally likely, the odds of any event can be found by dividing the outcome in question by all possible outcomes.  This told him the odds of rolling any number is 1/6 or around 17% of the time. So if we were to graph the odds of rolling any number it appears very uniform as follows.

Next, he made the leap to realize that dice have no memory – they are blind to outcome passed rolls. This more than anything else, is the point that most gamblers miss. He knew that rolling three sixes in a row didn’t not imply that another six was ‘due’. It also allowed him to realize that rolling two die at the same time, or rolling one dice twice was exactly the same event.

This idea offers a real advantage when he turned to the problem of calculating the odds of rolling any pair. Luckly for Cardano, with his concept of a probability space, could answer this question easily. Without even a pencil!

First we need all the possible outcomes when two dice are rolled. This is the same as rolling one dice two times, and it can result in 36 different outcomes. To figure out the odds of a pair he only needed to identify the number of ways you can roll a pair. There are  six ways to roll a pair (([1,1][2,2][3,3][4,4][5,5][6,6]), so the odds of rolling a pair is 6 divided by 36, or simply 1/6.

This is a simple, yet beautiful idea. To calculate the odds of an event, one needs to divide the number of ways it can happen by all possible outcomes. As we roll more dice, the possibility space grows exponentially. Just adding a third roll gives us 216 possibilities. So, what are the odds of rolling 3 ones? Easy, 1/216.

So, this chapter concludes with one sobering idea for a gambler. There are no lucky numbers or sequences – The best bet, is as good as a blind guess. If we roll a dice ten times, it’s just as likely to come up all ones as it is to come up any other sequence we can think of. It is only the probability space which grows when we try to predict many random events, and it is this idea which we will return 100′s of years later when we are desperately trying to hide our secrets.
Chapter 3. Needle in a Hay Stack: How wars are won

Now we return to our problem. Alice & Bob have split up, they need to communicate without detection…they need a secret language. This problem of secret language has been reoccurring since the birth of communication.

We return to 146BC. The Roman Empire comprised 100 million people, making it the largest and most powerful political empire in history, unsurpassed for over 1000 years. One of the greatest military leaders to shape it was Julius Caesar. During the Gallic Wars of 58 BC, Caesar changed the letters in his secret commands to make them unreadable should the enemy intercepted it. His method is now called the Caesar Cipher.

If Caesar needed to notify his lieutenant of the planned attack location of Gergovia. He would shift each letter in his message by some fixed number of positions in alphabet. With a shift of 3 he would map letters as follows, notice we wrap around at the end:

So his message of “GERGOVIA” would appear as “JHSTUJSYD” which seems to give no information about the original message. Altering a message to make it unreadable is called encryption. Incredibly, this simple form of encryption was strong enough to be used by military leaders for hundreds of years after Caesar! Its true weakness was only discovered 800 years later by an Arab mathematician named Al-Kindi. In the middle of the 9th Century, he published his Manuscript on Deciphering Cryptographic Messages. He attacks Caesars strategy using simple yet powerful ideas from statistics. He called his strategy frequency analysis, with frequency meaning how often something occurs. He knew that all languages use certain letters more often than others. Al-Kindi counted the letters in a large body of text to find out how often each occurs on average. If we do this with the modern English alphabet we find:

This graph captures the fingerprint of our language. Given a body of any English text, whether it is a book, newspaper or magazine, it will always exhibit this shape when graphed. Al-Kindi called this graph of letter probabilities the frequency distribution and it was a very powerful tool. It now became very easy to break Caesar’s cipher if one always begins by looking for the frequently occurring letters. For example, let’s say that, for some secret message, this turns out to be G and Y. Since the two most common letters in English are E and T one can check the graph and see the shift was likely 2.

E+2->G

T+2->Y

So the entire message can be easily discovered by shifting all letters back two places, this is called decryption. For example: CVVCEM – 222222 = ATTACK.

Once this method of attack became well known, it was clear that Caesar’s method must be improved. In 1553, an Italian named Bellaso was first to publish an idea to seriously improve Caesar’s method – now known as the Vigenère Cipher.  It works on a simple premise: Instead of shifting all letters by the same amount, make it more difficult by shifting each letter by a different amount. There are many ways to do this; one is to use a key word to define the shifts. So, instead of Alice and Bob agreeing on a secret shift number for their secret letter, they agree on a secret shift word. It would work like this:

Given a message: “THE MONEY IS IN THE PARK”

Chose some key word, such as a person’s initials “HAD”, and repeat it multiple times in order to match the message length -

THEMONEYISINTHEPARK + HADHADHADHADHADHADH

Each letter, is converted into a number based on its position in the alphabet to assign a number. Each number then defines the shifts as before.

THEMONEYISINTHEPARK + 8148148148148148148 = BIBVPRHJKDKDJHDJD

This method is more powerful precisely because it changes the frequency distribution of the message. Specifically, the graph is more uniform, common letters appear less frequently and less common letters appear more frequently. So the resulting code text will have a frequency graph similar to this:

It is difficult to identify the most common letters since many occur with similar frequency, the distribution has flattened. This requires extra time and guess work to decode the message, making Al-Kindi’s method much more troublesome. In fact, this was an incredibly secure form of encryption used in various forms for next 350 years.

Notice how the letter frequencies are even more uniform due to better mixing performed by the shifts – it’s almost impossible to determine the most frequently occurring letters since they all seem to occur with similar probability.

CUT TO Turn of the Century

-Turn of the 20th Century marked by the industrial revolution  (stock)

-leading to WW1? (stock)

-WW1 communication (stock)

-Idea emerged by Vernam (one-time pad) (pdf of patent)(Frank Miller banker who fought in the Civil war may have invented.)

- Alice Bob explanation

- Possibility space (stack of paper) (best enemy could do is guess!)

- WW2 begins. industry was “re-tooled” for war. split occurs.

- new problem, need fast way to generate random shifts continuously over time. Pads can run out. Move towards machines.

- Need “random shifting machines” (shot of adding machine spitting out random digits).

- Non-random keys would destroy security

-what are the powerful properties of a “true” random sequence?

-Light bulb example (coin vs guess) *uniform distribution at all scales*

-Light bulb example 2: (coin vs machine) *no repetition*

- Need a machine which rarely repeats and spits out a even mixture of shifts. However the machines are not actually random, the “random” element (as with dice) is the starting position.

“Similar probability” is behind the most important idea in this history of keeping secrets. In fact this idea was so powerful that the theory behind it was classified until 1949. In 1917 a man name Gilbert Vernam published his famous patent for the key ingredient needed to make secure codes: randomness. He realized that the secret shifts were greatly improved when defined randomly. The beauty of his idea lies in its simplicity.

His idea, called the “one-time pad” can be performed as follows: Alice’s message is 20 letters long. First, Alice and Bob must meet and generate 20 random shifts – this could be done using 20 rolls of an alphabet dice

Imagine the result of twenty rolls is: NLAGUWJNDPPEUNAWMM. This is their secret random key. The encryption and decryption work exactly as before, but now the shifts introduce the powerful behavior of randomness. When shifts are made according to random events, over time the distribution will show that each letter is equally likely:

Sharing long lists of random events to define shifts is a time consuming process. Initially this was done by sharing a list of random shifts on two identical pads of paper (hence the name one-time pad), however after each use the pad must be destroyed and never reused.

This became the key problem faced during World War 2.

[

Photos accessible on the Internet show captured KGB pads that fit in the palm of one's hand,^[4] or in a walnut shell.^[5] To increase security, one-time pads were sometimes printed onto sheets of highly flammable nitrocellulose.

]

By World War 2 communication secrecy was more important than ever. There was an explosion of technological developments in communication, like the telegraph and radio. More secret data flowed through the air and underneath the ground than ever before, and it all needed to be encrypted. By 1942 Germany had dominated central Europe, and Hitler’s war machine seemed nearly unstoppable.

The Allies (lead by Britain, United States and Soviet Union) had to break the encryption of the Axis powers (lead by German, Italy and Japan). This gave birth a highly secret intelligence project called Station X, employing more than 9,000 people during the course of the war. In fact, you could say that World War 2 was actually two battles: one a violent conflict on land, and the other waged by the world’s best Mathematicians and Engineers in secret offices.

During the War, the secrecy was loosely based on the one-time pad idea. However, there was a need to move away from paper lists and automate the process of generating random shifts. The solution was to build machines to generate the random shifts. Since the operation of machines is very predictable, engineers designed them with complicated shifting mechanisms in an attempt to simulate the unpredictability of random events. For this system would work it would require two parties to share a matching pair of ‘randomizing machines’.

The Germans worked on their own solution to guard their codes, and developed the Lorenz Cipher. It used a series of 12 rotors, chained together like the wheels on a car’s odometer. Instead of counting by one, a complicated arrangement of pins and cams output a stream of seemingly random values. The output of the Lorenz Cipher was fully predictable if you knew how the rotors were connected. This initial configuration can be thought of like the password to a combination lock. When two people wanted to communicate they would first share machine’s setup. They could then use the machines to automate the process of random shifts. A message would be fed into the machine, combined with the random values and then sent as a code. Next, the receiving machine would take in this code and output the original message:

To break this encryption the Allies needed mathematical geniuses with mechanical minds. Some of the most famous involved in this effort included Allan Turing, Claude Shannon and a Canadian named William Tutte. The real problem at the heart of the situation was the limit of a machine’s ability to generate randomness. The main area of attack was therefore not always the machines themselves but the behavior of the predictable human operators. They knew the best way to attack any encryption machine was to perform a plain-text attack. A plain-text attack occurred when the enemy intercepted a message both before and after its encryption. This code-message pair was known as a crib. Every crib gave away information about the shifts performed by the machines’ rotors. One crib that was especially devastating to the Germans occurred on August 30 1941. A German operator sent a 4,500 character message that was not received on the other end due an error. So, they asked for the message to be retransmitted using the same machine settings. The Allies intercepted the resent message.

From this code-message pair John Tillman recovered the 4500 character stream of random values produced by the Lorenz machine. Tillman gave this random stream of shifts to a Canadian mathematician, William Tutte. Tutte immediately noticed many repetitive patterns in the data, patterns that could not occur in true random events, that were the result of mechanical shifting. After studying the data he reverse engineered the design of the Lorenz shifting mechanism. This is widely considered to be the greatest achievement of World War 2 code breaking. This intelligence was incorporated into the first programmable computer named Colossus, which allowed the Allies to hasten the process of breaking German messages from weeks to hours.

During 1940, 178 German and Japanese messages were broken, nearly all successfully. The effort to successfully break the encryption helped the Allies defeat the Nazis two years earlier than expected. Germany surrendered on April 29th 1945,  but Japan refused. They faced the devastating results of the Manhattan project that followed close behind Station X. On August 6th and 9th the United States dropped atomic bombs on the Japanese cities of Hiroshima and Nagasaki – crushing Japan into defeat.

In 1945, an American engineer named Claude Shannon published a classified report called ‘A Mathematical Theory of Cryptography’ which proved that perfect secrecy could be achieved with the careful application of randomness. Most importantly, each random shift should never be reused. This was the mistake made by the Germans rendering their encryption scheme breakable. After Shannon, no one has, or ever will, change the definition of how to achieve perfect security.

In the years that followed this story took a final, unexpected twist, from war machines to bank machines.

Chapter 4. Messages without a meeting: The internet changes the game

At the end of World War 2 Europe was ravaged by war and two world superpowers emerged, Soviet Union and United States. Although these powers agreed on the need to dismantle the German political system, the question of what to do with Germany next was not as simple. In the following years Germany was divided into a western democratic government (funded mainly by the United States & Britain) and an eastern communist government lead by the Soviet Union. This division of German was the root of a larger race to maintain dominance over Western Europe, which would lead to a single world power. After United States tested and used the first atomic bomb, this need for dominance displayed itself as a competition to develop the biggest nuclear weapons arsenal. It spiraled out of control, and by 1965 the United States and Russia built more than 60,000 nuclear weapons.

As a result, the United States Defense Department designed a network called   Advanced Research Projects Agency Network, or ARPANET, which allowed them to communicate between weapon sites during an attack. It would also let them launch immediate retaliatory strikes from distant locations. ARPANET was the beginning of what we now call the Internet. It allows a web of computers, in distant locations, to communicate with each other. If one connection is destroyed, the message finds another pathway through the web. The system worked very well and was widely adopted by business and industry. By 1991, the Cold War ended and the entire world was connected by the Internet.

The data in messages transmitted around the world needed to be encrypted – especially money transfers. In order for two people to communicate in secrecy they both needed to share a secret random code. But how unless they met first, or sent the secret code by mail, which could be intercepted? This is the new security problem that the internet presents.

In 1976, Diffie and Hellman identified this problem in an innocent-looking paper titled “New Directions in Cryptography.” Until then, perfect encryption was thought to be impossible unless two people first shared a secret random key. Diffie and Hellman used an interesting analogy involving a lock to explain and solve the problem. Their theory is that some actions are easy performed, but not easily reversed. Think of a combination lock – it is easily locked if you have a key, but very difficult to open without one. Using a combination lock their solution would work as follows:

Recall that Alice is in hiding and needs to send Bob a secret.

1.  First Alice obtains a padlock, keeps the combination to herself, and sends it unlocked to Bob.

2. Bob generates random shifts (the random key) which will be used for one-time pad communication. He keeps one copy and puts the second copy into a box and locks it with Alice’s padlock and sends it back sealed.

3. Alice opens the padlock using her original combination and obtains the random key Bob created.

4. Now they can use the one-time pad scheme using the random key. They can do this without ever meeting, and be certain that no one else has the random key needed to read their encrypted messages unless they were able to guess Alice’s padlock combination.

To make this work in the digital world, we need the mathematical version of a combination lock. A mathematical lock is an operation on a number that is easy to perform, but difficult to reverse. For example, addition and subtraction do not work well as a lock, since adding to a number takes exactly as long as subtracting from it.

To find an operation which is difficult to undo we must return to Euclid’s original discovery and the problem of prime factorization – any number can be built by repeating some smaller prime numbers together.

To build up to C you need only the simple operation of multiplication A*B – this can be done using a calculator. But, given only C, it will take time to find the prime factor because it involves many divisions and checks. For example if we choose two prime numbers 79 and 41 to represent C it is very fast to perform 79*41.

However, anyone seeing only the number 3239 would not be able to find the prime factors as quickly. They would start testing from 2 to 3 to 4 to 5 until they hit 41, the first prime factor. So here we see a sharp difference between the time it takes to perform and operation A*B = C as compared to the reverse operation C = ?

Of course finding the prime factor of 3239 doesn’t take too long. But try to factor 18984748498940487374678930498. It’s possible, but it would take billions of steps. Using today’s super computers, finding the prime factor of a random 600 digit number would still take hundreds of years, but at this size multiplication is instantaneous by comparison.

Therefore the mathematical version of Alice’s padlock can be based on the difficulty of finding prime factors. This leads us to RSA encryption, published in 1978 by Ron Rivest, Adi Shamir, and Leonard Adleman. It is an encryption scheme based on the difficulty of finding the prime factors of a large number. Imagine Bob picks two  large prime numbers, A and B, then calculates C as AxB=C and share it with Alice. Alice can then shift her message in such a way that to decrypt it, would require the numbers A and B. Since only Bob knows A and B, if an enemy wanted to decrypt Alice’s message they must first determine the prime factors of C, which is impossible in a reasonable amount of time. It allowed the Diffie Hellman lock analogy to work, in which breaking open a lock is at least as difficult as finding the prime factors of C.

Incredibly, this trick of using mathematical locks based on difficult problems, solved the problem of Internet security. This allowed the information age to charge on with confidence. Today, no one worries about the secrecy of e-mails, bank transactions and messages – it’s all taken care of by careful use of prime numbers and randomness. This RSA algorithm runs on most computers in the world and is the most copied software in history. RSA takes Euclids prime factorization problem and uses it as the mathematical lock needed to perform the Diffie Helman secure key exchange.