## The Ultimate Shift Cipher

If you’ve seen the lesson on the one-time pad, you know that it is the **ultimate shift cipher**. It involves the application of a __random list of shifts__ equal to the __length of the message__. It’s important to understand exactly how and why the one-time pad is unbreakable, or, perfectly secret.

To understand why, we need to first introduce the **AND**, **OR** and **XOR **bitwise operations. Specifically why XOR must be used when performing the one-time pad on computers. **Bitwise** simply means that we are dealing with individual bits, or **binary numbers**. In any modern/computerized encryption scheme we represent our symbols using binary digits. If you forgot why, you can check out this video on Computer Memory

## Encrypting Colors

Let’s begin with a visual example by **encrypting a color** in the Khan Academy logo.

How do we turn a color into a number? Well, right now you are looking at HTML colors which are defined using the** RGB color model.** This is an additive model based on mixing some amount of **red, green and blue light**.

We can define exactly how much RED, GREEN and BLUE __using a number from 0-255__. Black is **all off** (0,0,0) while white is **all on** (255,255,255). In between there are **16 million possible colors** (256 * 256 * 256). Next let’s sample the green in the Khan Academy leaf in any image editing tool:

Notice it stores it as RED=156, GREEN=181 & BLUE=58. If we express these numbers in binary we get:

10011100 + 10110101 + 00111010. Which we can squeeze together as: 100111001011010100111010

**Binary RGB representation of the Khan Academy green:**

# Application of Random Shifts

Now let’s say you generate a shift sequence using coin flips converted into binary as:

**HTHTTHTHHHHTTHTTTTHTTHHH = 010110100001101111011000**

Let’s think about how we could **apply this shift sequence** to our color in order to encrypt it using the **one-time pad**

**100111001011010100111010 + 010110100001101111011000 = ?**

To make the one-time pad work we need to choose the correct operation so that the resulting sequence is equally likely to be any color. Let’s go over three different operations:

# AND

The **AND** operator is also known as **logical conjunction**, and works just like multiplication.

It outputs a 1 __only if all of the inputs are 1__. Here is the truth table:

**0** AND **0** = **0**

**0** AND **1** = **0**

**1** AND **0** = **0**

**1** AND **1** = **1**

** Let’s try it:** 100111001011010100111010 **AND** 010110100001101111011000 = **000110000001000100011000**

This results in a very dark purple. Notice when we perform the **AND** operation on any binary number, the resulting sequence **cannot be larger**. In our color example this eliminates many possible shades as it pushes the color **towards black**.

# OR

The **OR** operator is also known as **logical disjunction**. It outputs a 1 whenever __one or more of its inputs are 1__. Here is the truth table:

**0** OR **0** = **0**

**0** OR **1** = **1**

**1** OR **0** = **1**

**1** OR **1** = **1**

**Let’s try it:** 100111001011010100111010 **OR** 010110100001101111011000 = **110111101011111111111010**

This results in a light purple. Notice when we perform the **OR** operation on any binary sequence, the resulting sequence **cannot be smaller**. This eliminates many possibilities as it pushes the color **towards white**.

# XOR

The XOR operator outputs a 1 whenever the inputs __do not match__, which occurs when one of the two inputs is **exclusively true**. This is the same as addition mod 2. Here is the truth table:

**0** XOR **0** = **0**

**0** XOR** 1** = **1**

**1** XOR **0** = **1**

**1** XOR **1 **= **0**

**Let's try it:** 100111001011010100111010 **XOR** 010110100001101111011000 = 110001101010111011100010

This results in a slightly darker purple as compared to the OR operation. Notice when we perform the XOR operation on a binary sequence, **the resulting sequence could be any possible sequence**. Given some encrypted color, all we know is the original color is “equally likely to be any color”. We have __no information that could improve a blind guess__ (1/16 million).

Finally, let’s do a visual demonstration so that we can see the one-time pad in action. Then we can earn more energy points!

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