He rushed through the dingy streets, the sun bearing on his back, sweat flying as he hurried. A single arrow sliced through the air, just missing his ear by a few inches. But he was not so fortunate with the next two ones, one lodging itself in his leg, breaking his motion, the other piercing his arm. He was paralysed, trying to crawl through, but his adversary caught him in seconds. The shooter checked the man for the scroll, and found the one he was looking for, confirmed by the look of despair on the dying man's face. But as he opened it, he was bewildered, for the message made no sense, just some garbled alphabets here and there, probably in some foreign langauge altogether. But even upon investigating with the great literates of the nation, the language could not be figured out. In a far away cathedral, Caesar smiled.The message was secure.
Cryptography has been around us for ages, be it in the ancient Egyptian hieroglyphs or amongst the Hebrew scholars. One of the oldest instances of it, is the Caesar cipher,which was used by Julius Caesar to send messages of military importance across large distances, with them being reasonably secure upon inspection by third parties. Cryptography, as it's name suggests, is the study of techniques for secure communication in presence of third parties(or adversaries). The importance of this study came into major significance during the world wars where intercepting and decoding the enemies' messages could change the tide of the battle. This mostly led to rise of mechanical and electromechanical cipher machines, which were commonplace by World War II. In modern times, encryption is done through digital algorithms which have a key to encrpyt and decrypt messages. Claude Shannon,now considered by many to be the “Father of Mathematical Cryptography”, wrote that there are two systems of secrecy: First which secure communication against adversaries who are assumed to have infinite resources to decode a message ( Unconditional secrecy), and second to work against adversaries with a limited amount of resources ( Computational secrecy) .
An example of the first is the One-time Pad. In this technique, a plaintext is paired with random, secret key (or pad). Then, each bit or character of the plaintext is encrypted by combining it with the corresponding bit or character from the pad using modular addition. If the key is truly random, and at least as long as the plaintext, and never reused in whole or in part, and kept completely secret, then the resulting ciphertext will be impossible to decrypt or break. While being proven to being perfectly secure by Claude Shannon, there have been many practical drawbacks to using the One-time pad specially in times of twenty-first century with the advent of computers and algorithms which aren't that powerful but still more efficient for securing a large amount of messages. Another example is Shamir's Secret sharing which uses the idea that a polynomial of degree 't-1' can only be uniquely defined by 't' or more points passing through it. This can be used to allow secret sharing amongst a number of people. For example, say there exist 100 people and a dealer, who gives each of these 100 people a part of the secret. Through Shamir's secret sharing technique, one may be able to decipher the message if he/she knows the parts of a given amount of people, say 60 or more, that is, if any 60 of the 100 people come together, they can decode the secret, but any less would fail in the same. Not only secure,but the size of each part does not exceed the size of original data which makes it more efficient than many of it's contemporaries. Systems based on Computational secrecy deal with encryption processes based on some hardness assumptions. The very famous RSA cryptosystem which is used in a variety of places is based on the RSA problem, which deals with finding the eth roots of an arbitrary number, modulo N, where the factors of N are not known.
Turning onto a different aspect of cryptography, we talk about legal and privacy related issues. In today's world, threat analysis happens to be a very important part of national and global security, simply because of the potential damage of terrorism, et cetera. Thus cryptography has been of special interest to various intelligence agencies .Secret communications may be criminal or even treasonous, and thus some argue there should be some sort of monitoring and control over the private use of Cryptography. Many countries like Pakistan and Singapore have very tight restrictions on domestic use of Cryptography, specially because since the advent of the internet and computers, high quality encryption techniques have become easily available. Keeping this in mind, many nations have supported a legislation that requires individuals to surrender cryptographic keys to law enforcement, that is,the Key Disclosure Law. In countries like Australia, India, France and United Kingdom, an individual under investigation may be compelled to hand over his/her encryption key or passwords during a criminal investigation. At the same time there have been case in the United States of America, where a witness was protected from being forced to incriminate themselves. Thus,a fundamental question rises as to how much power should an individual have over his/her own privacy, while also factoring in the global welfare and necessities for one's own security, and whether in the name of national security, can we ask our own fellow countrymen to give up on their freedom.