Does calculus act as the pedestal of Cryptography?

What is cryptography?

Cryptography is a branch of computer science that involves the methods of protecting the data through the help of codes. It is closely associated with encryption, which is the act of scrambling ordinary text into what's known as ciphertext and then back again upon arrival. The communication of our computers is based on numbers, and – more importantly – fancy properties of numbers provide us with extremely ingenious cryptosystems. Cryptography is the secure information and communication techniques derived from mathematical concepts and a set of rule-based calculations called algorithms, to transform messages in ways that are hard to decipher.

History of Mathematics in Cryptography

Modern cryptography mathematics is much different than in the old days, it is still based on the same concepts used in ancient times. The earliest known instances of cryptography were carved in the Old Kingdom of Egypt, dating back to 1900 BCE. There were clay tablets from Mesopotamia, that were to have encryption to hide a recipe for pottery glaze since it was likely for commercial value and private to the seller.

Cryptography has advanced considerably in the past 100 years. You have likely seen simple cryptography, such as the Caesar cipher, also known as the shift cipher. ROT13 is one of the most common types of shift ciphers, which means that the alphabet rotates “13” times. See this figure below:

However, this is not the type of mathematical cryptography that can protect data in today’s world. The latest technology is used mostly to protect personal and sensitive financial information so that it is safely transmitted online.

Calculus: The Prophet

It is a proverb: ‘Tell me who your relatives are and I will tell who you are’. The calculus says: ‘Tell me who your neighbors are and I will tell you who you are’. Calculus shows a particular trend, pattern, and regularity. For example, the Taylor series tries to predict the value of a function by just knowing its neighborhood. The value of a differentiable function at a+b is: f(a+b) = f(a) + bf’(a). This nature of calculus is prevalent in every field of the physical world.

Calculus in encryption

[1] The thing of lament is that calculus cannot meet the demand of being so precise and careful at a particular time. When working on a continuous and differentiable domain, there are high chances of getting errors due to truncation. Calculus has a major handicap in using its various function for encrypting data. If two input values are close, then by using the concept of continuity, the output values will be close too. One can easily predict the encrypted data on the basis of any leak of partial information. Encryption is meant to misguide an individual who tries to decrypt the data. The mathematical numbers are treated as quantities in the physical world and are then approximated. The assumptions of what these numbers signify in the sector of cryptography are pole apart from what they actually mean in the real world. Thus, calculus doesn’t play any significant role in cryptography due to the difference in the way the numbers are looked upon in calculus and cryptography.

References:

[1] P. Vanchinathan, “ Is calculus a failure in cryptography ?”, Resonance, 21,  239-245 ( 2016)

[2] WWW.PrivacyCanada.com