Sunday, April 21, 2019

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**Scott Lewis****Member**- From:: Richmond, VA
- Registered: 2003-06-10
- Posts: 58
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In 1977, three students from MIT (Massachusetts Institute of Technology), devised a number that was 129 digits long, that is the product of two prime numbers, meaning that there were only two non-trivial factors of this number. The students were Ronald Rivest, Adi Shamir and Leonard Adleman. The number came to be known as RSA129. They used this number and the fact that it had only two factors other than 1 and the number itself as a method of encryption. This method established Rivest, Shamir and Adleman as the leaders in the field of cryptography.

RSA129 was published in the August 1977 issue of Scientific American along with a challenge that if anyone could factor the number and decipher their hidden message, they would win a prize of $100. It was commented by one authority that the money would be safe for the next 20,000 years. It, in fact, was safe for only 17 years. A team of 600 mathematicians factored the number over a period of 8 months. The two prime factors of the 129 digit number were respectively 64 and 65 digits long.

This technique and the extreme difficulty of factoring numbers larger than 100 digits is the basis of modern crypto systems. Most modern systems use a number of 200 digits raised to a power of a number of 100 digits and greater. Now that, my friends, is some serious math. However, this sytem, in theory, can still be broken. The idea is not that any given system is impossible to crack but rather that the time required to crack it is so great that by the time it is cracked, the information is useless. Some estimates say that factoring a number of 400 digits using a desktop computer would take **billions of years**.

It may be that I love this subject because I have always secretly dreamed of being a James Bond type of secret agent, but encryption is one of my favorite subjects and this month that is what we are going to talk about. I must first qualify this article and any statements made herein by confessing that I am a novice at encryption. Many of you may, and probably do, have much more experience and expertise in this area than I.

While there is no such thing as a 100% secure encryption method, the particular method I will talk about is very basic and the security of it is minimal. It should be accepted for introductory and educational purposes only. If you want to be able to send semi-sensitive emails that can only be read by someone with the "key" to decipher the message, this will suffice. **If you are sending sensitive information, like credit card numbers or financial statements you definitely should not use this method**.

The average person is not going to be able to crack messages encrypted with this formula but someone with even rudimentary knowledge of code breaking probably can so **use this method at your own risk**.

Before getting into the actual code it is necessary to define some basic terms that will be used frequently in this article. An understanding of the two basic encryption methods will also be helpful.

As mentioned, there are two basic forms of encryption: transposition and substitution. Almost everyone is familiar with both methods, though they may not realize it. Anyone who has ever played a word scramble puzzle is familiar with transposition. Likewise, anyone who has ever played with morse code has used substitution.

Transposition is a rearrangement of the characters in the message itself. It is the least secure of the two types. One example of encryption of the phrase "The eagle has landed" by the transposition method might produce "het "gleea sha dleand". Transposition can and normally is much more complex than this example, using geometric patterns and transposition of blocks of the text.

In substitution, on letter of the alphabet is substituted for another. An example of the substitution method is shown in the diagram below.**================================a b c d e f g h i j k l m n o p q r s t u v w x y zd e f g h i j k l m n o p q r s t u v w x y z a b c================================**

So our message "The eagle has landed" encrypted with the substitution scheme above is "wkh hdjoh kdv odrghg".

Keep in mind, that any encryption method needs to be systematic so that once encrypted, there exists some way of unscrambling the message quickly and accurately. To achieve this I will use the following formulas:

For enciphering:

For deciphering:

So, the in the inverse

It is also going to be important to define a couple of mathematical terms to help us with encryption. They are

The concept of

Multiplicative inverse is also essential to successfully enciphering and deciphering our message because in unscrambling the message and arriving back at the starting point, we will need to know how to get from say

To get around this, it is necessary to find a number by which we can multiply our

Okay, so now we're ready to jump in with some AppleScript code. I know what you're thinking,

## Applescript:

on getInverse(m, n)

set m to m as integer

set m2 to 0

repeat until ((m * m2) mod n) = 1

set m2 to m2 + 1

end repeat

return m2

end getInverse

If we call this subroutine, substituting **5** for **m** and **26** for **n**, the result will be **21**, because **5*21 mod 26 = 1**.

Okay, now we have a formula for finding our enciphering and deciphering keys right Well, not exactly. There is a problem I did not warn you about. That is that the value of **m** must be **relatively prime** to the **modulus**. Two numbers are said to be **relatively prime when the only factor they have in common is 1**. This is crucial because if we use a value of say **12** with a **modulus of 26**, whose common factors are 1 and 2, we will not have a one-to-one ratio of characters between our **ciphertext** and **plaintext**. This means that there will be more than one possible decription of our message. Our "key finding formula" will not work either because there is no number by which you can multiply 12 that when lowered modulus 26 will yield 1. I won't go into the math behind all of this so I'm going to have to ask you to trust me. You can, if you like, try using the routine above with **m=12** and **n=26** but you will be waiting for a very, very long time for the results.

Okay, so now we know that **any number, relatively prime to our modulus is a valid value for (m)**.

We are ready to introduce some simple code for actually encrypting a message. I will first give the formula, then the code.

The formula is **C=m*P mod n**. And, as promised here's the code:**]Enciphering routine**

## Applescript:

on encipherText(p, m, n)

set c to (p * m) mod n

return c

end encipherText

Let's test it using the letter **h** whose position in our set of 26 characters is **8**.**]Encipher the character "h"**

## Applescript:

set c to encipherText(8, 5, 26)

The result is **14**. So our plaintext character **h** becomes our ciphertext **n** because **n** is the **14**th character in our 26 character set. Isn't this exciting!

You might well ask "That's great but how do I get back to the plaintext character". We have already established the tools we need to achieve this.

We can use our subroutine for finding multiplicative inverses above and the formula for deciphering our ciphertext. The formula is **P=(m*C) mod n**.

Below is the code for this formula. You might notice that it is exactly the same as the encriphering code. I prefer to use two separate routines in order to keep the steps clear in my mind, but in fact, you can use the same routine for both.**]Deciphering routine**

## Applescript:

on decipherText(c,m,n)

set p to (c * m) mod n

return

end decipherText

So we run the following code:**]Get our Decrkption key (kD)**

## Applescript:

set kd to getInverse(5, 26)

Which results in **21**. Now all we do is plug our values into our deciphering routine:**Decipher the character "v"**

## Applescript:

set p to decipherText(14, 21, 26)

This returns a value for **p** of **8**, which in turn is the numerical value of the letter **h** in our 26 character set. So we have arrived back where we started.

A reasonable man adapts himself to the world he sees. An unreasonable man tries to adapt the world he sees to himself. Therefore, all advancement in society is achieved by unreasonable men. --Paraphrase of a quote by George Bernard Shaw

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