Lets Learn About XOR Encryption

One of the more common things about this generation is the constant desire to write up (type) their thoughts. So many of the conversations from my high school days were long lasting, but quickly forgotten. Today’s generation is much more likely to blog, tweet, write status updates or simply open up a notepad file and write up their thoughts after such a conversation.

When we feel that our thoughts are not ready for public eyes (maybe you want to run your idea by the Patent and Trademark Office before speaking about it) we may seek some form of security to ensure that they stay private. An old fashioned way of doing this was to write in a diary and enclosed it within a lock and key. The mathematical field of encryption also tries to grant privacy by encoding messages so that only people with the necessary information can read them.

The type of encryption I want to speak about today is called XOR encryption. It is based on the logical operation called “exclusive or” (hence the name XOR). The exclusive or operation is true between two logical statements if exactly one of the two statements is true, but not both statement. This can be represented with the following truth table

Input 1 Input2 XOR Result
T T F
T F T
F T T
F F F

XOR Encryption is particularly useful in this day and age because we every character we type is understood by the computer as a sequence of zeros and ones. The current standard encoding that is used is Unicode (also known as UTF-8). Under this encoding the letter ‘a’ is represented as the binary string ‘01100001’. Similarly every letter, number and special character can be represented as its own binary string. These binary strings are just an assignment of numbers to these characters so that we can to help represent them in the computer. The numbers can the be thought of in base 10, which is how we generally think about numbers, or in base 2 which is how computers generally work with numbers (or a number of other ways). The way we would use these binary strings in encoding is first by translating a text from human-readable text to machine readable text via its binary string. For example, the word “Invincible”, we would get the following binary strings:

Letter Unicode in base 10 Unicode in base 2
I 73 01001001
n 110 01101110
v 118 01110110
i 105 01101001
n 110 01101110
c 99 01100011
i 105 01101001
b 98 01100010
l 108 01101100
e 101 01100101

To encrypt the message we need a key to encode the message and will simply perform an XOR operation on the key and every character in the string. Similarly, do decrypt the message we perform XOR operation on the key and every character in the encoded message. This means that the key (much like a normal key to a diary) must be kept private and only those whom the message is to be shared between have access to it.

Here is a link to the script where you can check out XOR Encrpytio. Try it out and let me know what you think.

Discrete-time Markov Chains

Much of how we interact with life could be described as transitions between states. These states could be weather conditions (whether we are in a state of “sunny” or “rainy”), the places we may visit (maybe “school”, “the mall”, “the park” and “home”), our moods (“happy”, “angry”, “sad”). There are a number of other ways to model states and even the possibility of infinitely many states.

Markov Chains are based on the principle that the future is only dependent on the immediate past. so for example, if I wished to predict tomorrow’s weather using a Markov Chain, I would need to only look at the weather for today, and can ignore all previous data. I would then compare the state of weather for today with historically how weather has changed in between states to determine the most likely next state (i.e what the weather will be like tomorrow). This greatly simplifies the construction of models.

To use Markov Chains to predict the future, we first need to compute a transition matrix which shows the probability (or frequency) that we will travel from one state to another based on how often we have done so historically. This transition matrix can be calculated by looking at each element of the history as an instance of a discrete state, counting the number of times each transition occurs and dividing each result by the number of times the origin state occurs. I’ll next give an example and then I’ll focus on explaining the Finite Discrete State Markov Chain tool I built using javascript.

Next, I want to consider an example of using Markov Chains to predict the weather for tomorrow. Suppose that we have observed the weather for the last two weeks. We could then use that data to build a model to predict tomorrow’s weather. To do this, lets first consider some states of weather. Suppose that a day can be classified in one of four different ways: {Sunny, Cloudy, Windy, Rainy}. Further, suppose that over the last two weeks we have seen the following pattern.

Day 1 Sunny
Day 2 Sunny
Day 3 Cloudy
Day 4 Rain
Day 5 Sunny
Day 6 Windy
Day 7 Rain
Day 8 Windy
Day 9 Rain
Day 10 Cloudy
Day 11 Windy
Day 12 Windy
Day 13 Windy
Day 14 Cloudy

We can look at this data and calculate the probability that we will transition from each state to each other state, which we see below:

Rain Cloudy Windy Sunny
Rain 0 1/3 1/3 1/3
Cloudy 1/2 0 1/2 0
Windy 2/5 1/5 2/5 0
Sunny 0 1/3 1/3 1/3

Given that the weather for today is cloudy, we can look at the transition matrix and see that historically the days that followed a cloudy day have been Rainy and Windy days each with probability of 1/5. We can see this more mathematically by multiplying the current state vector (cloudy) [0, 1, 0, 0] by the above matrix, where we obtain the result [1/2, 0, 1/2, 0].

In similar fashion, we could use this transition matrix (lets call it T) to predict the weather a number of days in the future by looking at Tn. For example, if we wanted to predict the weather two days in the future, we could begin with the state vector [1/2, 0, 1/2, 0] and multiply it by the matrix T to obtain [1/5, 4/15, 11/30, 1/6].

We can also obtain this by looknig at the original state vector [0, 1, 0, 0] and multiplying it by T2.

T2 =

1

3/10 8/45 37/90 1/9
1/5 4/15 11/30 1/6
13/50 16/75 59/150 2/15
3/10 8/45 37/90 1/9

When we multiply the original state vector by T2 we arrive at this same answer [1/5, 4/15, 11/30, 1/6]. This matrix T2 has an important property in that every state can reach every other state.

In general, if we have a transition matrix where for every cell in row i and column j, there is some power of the transition matrix such that the cell (i, j) in that matrx is nonzero, then we say that every state is reachable from every other state and we call the Markov Chain regular.

Regular Markov Chains are important because they converge to what’s called a steady state. These are state vectrs x = [x0, …, xn] such that xTn = x for very large values of n. The steady state tells us how the Markov Chain will perform over long periods of time. We can use algebra and systems of linear equations to solve for this steady state vector.

For the Javascript program I’ve written, I have generated a set of painting samples for a fictional artist. The states are the different colors and the transitions are the colors that the artist will use after other colors. as well as the starting and ending colors. Given this input, we can form a Markov Chain to understand the artist’s behavior. This Markov Chain can then be used to solve for the steady state vector or to generate random paintings according to the artist’s profile. Be sure to check it out and let me know what you think.