Encoded Math
Humans have been encoding messages for thousands of years to either protect sensitive information or provide self-amusement. Each code, however, only works if there is a consistent pattern that can allow another person to decipher it. For example, take the Caesar shift code. Every letter is shifted down a certain amount of spaces so that it is easy to write and decode a message.
However, with advanced computing, simple codes are easily broken and information can be stolen. This led to the advancement of the field of cryptology. Computers now encode information with the help of large primes, which I will further explain in this post!
For any privacy system, data must be “encrypted“ and then “decrypted“ to once again view the data. This is done with a “key“ thatThere are a few different mathematical terms and sequences in cryptology listed by “Privacy Canada.“ They are explained below:
Big O Notation:
This is in reference to any type of algorithm. It analyzes the complexity of an algorithm as some function. The most common example is O(n^2) Big 0 Squred. This is a type of “selection sort algorithm,” described by “Free Code Camp“ as something that iterates through a list to “ensure every element at inde i is the ith smallest/largest element.“ In otherwords, the input is the size of the list and the algorithm continues running for (n-1) times until it only runes onces. This means that the amount of repeated is a geometric function. The total “runs“ is equal to (1 + 2 + … + n) or:
Big O Notation only looks at the leading, or highest term, which is n^2. Because the way an algorithm grows increases “significant digits“ beyond what the initial value starts with, Big O Notation is justified in only looking at the dominant value.
General rules with growth are that exponentials are more complex than polynomials, factorials are more complex than exponentials, and multiplying terms will never be more complex than the next greatest power of n.
The complexity factor, because it is related to the number of times an algorithm will have to repeat, can then be used to determine calculation/run time. This in turn related to the amount of time to “decrypt“ a code based on an encryption/decryption key. While complexity might be good for data someone wishes to store, recovering that data regularly is extremely inefficient.
The next most obviously mathematical tangent encryption algorithm is Prime Factorization.
One type of this method uses semi-prime numbers, or the product of two prime numbers. in order to factor any number, you have to divide by different primes that are smaller than it until all you are left with is the multiples of primes. Prime numbers themselves are infinite, so there will never be a limit of un-used primes to chose from. This is challenging for larger numbers since every prime number below it will have to be tested, taking a large amount of time. The “public“ encrypted might be know, but the “key“ is knowing the factorization of the chosen large prime or semi-prime number.
In cryptology, this type of mathematical operation is called a “trap door.” Is is easy for a computer to just multiply two numbers together, but it requires substantially more time and operations to discover that new number’s factors. “The science of encryption: prime numbers and mod n arithmetic“ illustrates decoding a public key very well. The link can be found in sources.
Each type of encryption tool starts relatively simple, but just like exponential functions, increases in complexity rapidly. Next time you set a passwords or log on to a secure website, take into account how much math and time goes into keeping that data secure to just you.
Sources:
Guide to cryptography mathematics. (2022, July 05). Retrieved November 2, 2022, from https://privacycanada.net/mathematics/#:~:text=Cryptography%20Mathematics%20%E2%80%93%20This%20refers%20to,original%20text%20from%20encrypted%20keys.
Guide to cryptography mathematics. (2022, July 05). Retrieved November 2, 2022, from https://privacycanada.net/mathematics/#:~:text=Cryptography%20Mathematics%20%E2%80%93%20This%20refers%20to,original%20text%20from%20encrypted%20keys.
The science of encryption: Prime numbers and mod arithmetic. (n.d.). Retrieved November 6, 2022, from https://math.berkeley.edu/~kpmann/encryption.pdf