What is the term used to describe a cryptographic method that incorporates mathematical operations?

Cryptology is the mathematics, such as number theory and the application of formulas and algorithms, that underpin cryptography and cryptanalysis. Cryptanalysis concepts are highly specialized and complex, so this discussion will concentrate on some of the key mathematical concepts behind cryptography, as well as modern examples of its use.

In order for data to be secured for storage or transmission, it must be transformed in such a manner that it would be difficult for an unauthorized individual to be able to discover its true meaning. To do this, security systems and software use certain mathematical equations that are very difficult to solve unless strict criteria are met. The level of difficulty of solving a given equation is known as its intractability. These equations form the basis of cryptography.

Types of cryptology equations

Some of the most important equations used in cryptology include the following.

The discrete logarithm problem

The best way to describe this problem is first to show how its inverse concept works. Assume we have a prime number, P (a number that is not divisible except by 1 and itself). This P is a large prime number of over 300 digits. Let us now assume we have two other integers, a and b. Now, say we want to find the value of N, so that value is found by the following formula:

N = ab mod P, where 0 ≤ N ≤ (P · 1)

This is known as discrete exponentiation and is quite simple to compute. However, the opposite is true when we invert it. If we are given P, a, and N and are required to find b so that the equation is valid, then we face a tremendous level of difficulty. This problem forms the basis for a number of public key infrastructure (PKI) algorithms, such as Diffie-Hellman and EIGamal.

The integer factorization problem

This is simple in concept. Say, someone takes two prime numbers, P2 and P1, which are both "large" (a relative term, the definition of which continues to move forward as computing power increases). We then multiply these two primes to produce the product, N. The difficulty arises when, being given N, we try to find the original P1 and P2. The Rivest-Shamir-Adleman PKI encryption protocol is one of many based on this problem. To simplify matters to a great degree, the N product is the public key, and the P1 and P2 numbers are, together, the private key.

The elliptic curve discrete logarithm problem

This is a cryptographic protocol based upon a reasonably well-known mathematical problem. Mathematicians have studied the properties of elliptic curves for centuries but only began applying them to the field of cryptography with the development of widespread computerized encryption in the 1970s.

First, imagine a huge piece of paper, on which is printed a series of vertical and horizontal lines. Each line represents an integer, with the vertical lines forming x class components and horizontal lines forming the y class components. The intersection of a horizontal and vertical line gives a set of coordinates (x,y). In the highly simplified example below, we have an elliptic curve that is defined by the equation:

y2 + y = x3 · x2

For the above, given a definable operator, we can determine any third point on the curve given any two other points. This definable operator forms a "group" of finite length. To add two points on an elliptic curve, we first need to understand that any straight line that passes through this curve intersects it at precisely three points. If we define two of these points as u and v, we can then draw a straight line through these points to find another intersecting point at w. We can then draw a vertical line through w to find the final intersecting point at x. Now, we can see that u + v = x. This rule works when we define another imaginary point, the origin, or O, which exists at theoretically extreme points on the curve. The problem appears to be quite intractable, requiring a shorter key length (thus, allowing for quicker processing time) for equivalent security levels as compared to the integer factorization problem and the discrete logarithm problem.

Modern cryptology examples

Today, researchers use cryptology as the basis for encryption in cybersecurity products and systems that protect data and communications. A few examples of modern applications include the following.

Symmetric-key cryptography. Symmetric-key cryptography, sometimes referred to as secret-key cryptography, uses the same key to encrypt and decrypt data. Encryption and decryption are inverse operations, meaning the same key can be used for both steps. Symmetric-key cryptography's most common form is a shared secret system, in which two parties have a shared piece of information, such as a password or passphrase, that they use as a key to encrypt and decrypt information to send to each other.

Public-key cryptography. Public-key cryptography is a cryptographic application that involves two separate keys -- one private and one public. While both keys are mathematically related to one another, only the public key can be used to decrypt what has been encrypted with the private key. The most well-known application of public-key cryptography is for digital signatures, which allow users to prove the authenticity of digital messages and documents. It also makes it possible to establish secure communications over insecure channels.

Cryptanalysis. Cryptanalysis is the practice of analyzing cryptographic systems in order to find flaws and vulnerabilities. For example, cryptanalysts attempt to decrypt ciphertexts without knowledge of the encryption key or algorithm used for encryption. Cryptanalysts use their research results to help to improve and strengthen or replace flawed algorithms.

Cryptographic primitives. A cryptographic primitive in cryptography is a basic cryptographic technique, such as a cipher or hash function, used to construct subsequent cryptographic protocols. In a common scenario, a cryptographic protocol begins by using some basic cryptographic primitives to construct a cryptographic system that is more efficient and secure.

Cryptosystems. Cryptosystems are systems used to encode and decode sensitive information. Cryptosystems incorporate algorithms for key generation, encryption and decryption techniques to keep data secure. The basic principle of a cryptosystem is the use of a ciphertext to transform data held in plaintext into an encrypted message.

Cryptography is a method of protecting information and communications through the use of codes, so that only those for whom the information is intended can read and process it.

In computer science, cryptography refers to 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. These deterministic algorithms are used for cryptographic key generation, digital signing, verification to protect data privacy, web browsing on the internet and confidential communications such as credit card transactions and email.

Cryptography techniques

Cryptography is closely related to the disciplines of cryptology and cryptanalysis. It includes techniques such as microdots, merging words with images and other ways to hide information in storage or transit. However, in today's computer-centric world, cryptography is most often associated with scrambling plaintext (ordinary text, sometimes referred to as cleartext) into ciphertext (a process called encryption), then back again (known as decryption). Individuals who practice this field are known as cryptographers.

Modern cryptography concerns itself with the following four objectives:

1. Confidentiality. The information cannot be understood by anyone for whom it was unintended.
2. Integrity.The information cannot be altered in storage or transit between sender and intended receiver without the alteration being detected.
3. Non-repudiation. The creator/sender of the information cannot deny at a later stage their intentions in the creation or transmission of the information.
4. Authentication. The sender and receiver can confirm each other's identity and the origin/destination of the information.

Procedures and protocols that meet some or all of the above criteria are known as cryptosystems. Cryptosystems are often thought to refer only to mathematical procedures and computer programs; however, they also include the regulation of human behavior, such as choosing hard-to-guess passwords, logging off unused systems and not discussing sensitive procedures with outsiders.

Cryptographic algorithms

Cryptosystems use a set of procedures known as cryptographic algorithms, or ciphers, to encrypt and decrypt messages to secure communications among computer systems, devices and applications.

A cipher suite uses one algorithm for encryption, another algorithm for message authentication and another for key exchange. This process, embedded in protocols and written in software that runs on operating systems (OSes) and networked computer systems, involves:

• public and private key generation for data encryption/decryption
• digital signing and verification for message authentication
• key exchange

Types of cryptography

Single-key or symmetric-key encryption algorithms create a fixed length of bits known as a block cipher with a secret key that the creator/sender uses to encipher data (encryption) and the receiver uses to decipher it. One example of symmetric-key cryptography is the Advanced Encryption Standard (AES). AES is a specification established in November 2001 by the National Institute of Standards and Technology (NIST) as a Federal Information Processing Standard (FIPS 197) to protect sensitive information. The standard is mandated by the U.S. government and widely used in the private sector.

In June 2003, AES was approved by the U.S. government for classified information. It is a royalty-free specification implemented in software and hardware worldwide. AES is the successor to the Data Encryption Standard (DES) and DES3. It uses longer key lengths -- 128-bit, 192-bit, 256-bit -- to prevent brute force and other attacks.

Public-key or asymmetric-key encryption algorithms use a pair of keys, a public key associated with the creator/sender for encrypting messages and a private key that only the originator knows (unless it is exposed or they decide to share it) for decrypting that information.

Examples of public-key cryptography include:

• RSA, used widely on the internet
• Elliptic Curve Digital Signature Algorithm (ECDSA) used by Bitcoin
• Digital Signature Algorithm (DSA) adopted as a Federal Information Processing Standard for digital signatures by NIST in FIPS 186-4
• Diffie-Hellman key exchange

To maintain data integrity in cryptography, hash functions, which return a deterministic output from an input value, are used to map data to a fixed data size. Types of cryptographic hash functions include SHA-1 (Secure Hash Algorithm 1), SHA-2 and SHA-3.

Cryptography concerns

Attackers can bypass cryptography, hack into computers that are responsible for data encryption and decryption, and exploit weak implementations, such as the use of default keys. However, cryptography makes it harder for attackers to access messages and data protected by encryption algorithms.

Growing concerns about the processing power of quantum computing to break current cryptography encryption standards led NIST to put out a call for papers among the mathematical and science community in 2016 for new public key cryptography standards.

Unlike today's computer systems, quantum computing uses quantum bits (qubits) that can represent both 0s and 1s, and therefore perform two calculations at once. While a large-scale quantum computer may not be built in the next decade, the existing infrastructure requires standardization of publicly known and understood algorithms that offer a secure approach, according to NIST. The deadline for submissions was in November 2017, analysis of the proposals is expected to take three to five years.

History of cryptography

The word "cryptography" is derived from the Greek kryptos, meaning hidden.

The prefix "crypt-" means "hidden" or "vault," and the suffix "-graphy" stands for "writing."

The origin of cryptography is usually dated from about 2000 B.C., with the Egyptian practice of hieroglyphics. These consisted of complex pictograms, the full meaning of which was only known to an elite few.

The first known use of a modern cipher was by Julius Caesar (100 B.C. to 44 B.C.), who did not trust his messengers when communicating with his governors and officers. For this reason, he created a system in which each character in his messages was replaced by a character three positions ahead of it in the Roman alphabet.

In recent times, cryptography has turned into a battleground of some of the world's best mathematicians and computer scientists. The ability to securely store and transfer sensitive information has proved a critical factor in success in war and business.

Because governments do not want certain entities in and out of their countries to have access to ways to receive and send hidden information that may be a threat to national interests, cryptography has been subject to various restrictions in many countries, ranging from limitations of the usage and export of software to the public dissemination of mathematical concepts that could be used to develop cryptosystems.

However, the internet has allowed the spread of powerful programs and, more importantly, the underlying techniques of cryptography, so that today many of the most advanced cryptosystems and ideas are now in the public domain.