Cryptography, Encryption, and Number Theory

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Cryptography, Encryption, and Number Theory


Cryptography is a branch of applied mathematics concerned with developing codes to enhance the privacy of communications. It is equally concerned with methods for breaking codes. When successful, cryptography allows its users, whether governments, military, businesses, or individuals, to maintain privacy and confidentiality in their communications.

Encryption is any form of coding, ciphering, or secret writing. Encryption of data, therefore, includes any and all attempts to conceal, scramble, encode, or encipher any information. In the modern world, however, the term data usually implies digital data, that is, information in the form of binary digits (bits, most often symbolized as 1s and 0s).

The goal of encryption is to becrack proof(i.e, messages should only able to be decoded and understood by authorized recipients). Cryptography is also a means to ensure the integrity and preservation of data from tampering. Modern cryptographic systems rely on functions associated with advanced mathematics, including the branch of mathematics known as number theory, which explores the properties of numbers and the relationships between numbers.

Although cryptography has a long history of use in military and diplomatic affairs, its importance increased greatly during the later half of the twentieth century. Growing reliance on electronic communication and data storage increased demand for advancements in cryptologic science. The use of cryptography broadened from its core diplomatic and military users to become of routine use by companies and individuals seeking privacy in their communications.

In addition to improvements made to cryptologic systems based on information made public from classified government research programs, international scientific research organizations devoted exclusively to the advancement of cryptography (e.g., the International Association for Cryptologic Research), began to apply applications of mathematical number theory to enhance privacy, confidentiality, and the security of data. Number theory was applied to develop increasingly involved algorithms (step-by-step procedures for solving a mathematical problems). In addition, as commercial and personal use of the Internet grew, it became increasingly important not only to keep information secret but also to be able to verify the identity of message sender. Cryptographic use of certain types of algorithms calledkeysallow information to be restricted to a specific and limited audiences whose identities can be authenticated.

In some cryptologic systems, encryption is accomplished by choosing certain prime numbers and then products of those prime numbers as basis for further mathematical operations. In addition to developing such mathematical keys, the data itself is divided into blocks of specific and limited length so that the information that can be obtained even from the form of the message is limited. Decryption is usually accomplished by following an elaborate reconstruction process that itself involves unique mathematical operations. In other cases, decryption is accomplished by performing the inverse mathematical operations performed during encryption.

In August, 1977, Ronald Rivest, Adi Shamir, and Leonard Adleman published an algorithm destined to become a major advancement in cryptology. (It is often debated whether the RSA algorithm was, at least it part, developed earlier and independently by specialists working for the British intelligence agencies.) The RSA algorithm underlying the system derives its security from the difficulty in factoring very large composite numbers. With the rise of the Internet, the RSA algorithm became the most commonly used encryption and authentication algorithm in the world. It has been used in the development of Internet web browsers, spreadsheets, data analysis, email, and word processing programs. Every time a credit-card purchase is made on a Web site, for example, the RSA algorithm is used by encrypt the buyers credit-card number for transmission to the seller.

Because digital data are numerical, their efficient encryption demands the use of ciphering rather than coding. A code may be purely verbal, but a cipher is a system of rules for transforming any message text (the plaintext) into an apparently random text (the ciphertext) and back again. Digital computers are ideal for implementing ciphers; virtually all ciphering today is performed on digital data by digital computers.

See also Computer languages; Computer memory, physical and virtual memory; Computer software; Internet and the World Wide Web.



Koblitz, Neal, et al. Algebraic Aspects of Cryptography. New York: Springer, 2004.

Lehtinen, Rick, et al. Computer Security Basics. Sebastopol, CA: OReilly Media, 2006.

Pincock, Stephen. Codebreaker: The History of Secret Communications. Walker & Company, 2006.


National Institute of Standards and Technology.Advanced Encryption Standard: Questions and Answers.Computer Resource Security Center. March 5, 2001. <> (accessed October 24, 2006).

Nechvatal, James, et al.Report on the Development of the Advanced Encryption Standard.National Institute of Standards and Technology. October 2, 2000. <> (accessed October 24, 2006).

K. Lee Lerner

Larry Gilman