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Cryptography or cryptology is the science or art of message secrecy. In modern times, cryptography is considered to be a branch of both mathematics and computer science, and is affiliated closely with information theory, computer security, and engineering. Cryptography is used in many applications encountered in everyday life; examples include security of automated teller machine cards, computer passwords, and electronic commerce all depend on cryptography.

Symmetric-Key Cryptography

Symmetric-key cryptography refers to encryption methods in which both the sender and receiver share the same key (or, less commonly, in which their keys are different, but related in an easily computable way). This was the only kind of encryption publicly known until 1976.

The modern study of symmetric-key ciphers relates mainly to the study of block ciphers and stream ciphers and to their applications. A block cipher is, in a sense, a modern embodiment of Alberti's polyalphabetic cipher: block ciphers take as input a block of plaintext and a key, and output a block of ciphertext of the same size. Since messages are almost always longer than a single block, some method of knitting together successive blocks is required. Several have been developed, some with better security in one aspect or another than others. They are the mode of operations and must be carefully considered when using a block cipher in a cryptosystem.

  • Stream ciphers, in contrast to the 'block' type, create an arbitrarily long stream of key material, which is combined with the plaintext bit-by-bit or character-by-character, somewhat like the one-time pad. In a stream cipher, the output stream is created based on an internal state which changes as the cipher operates. That state's change is controlled by the key, and, in some stream ciphers, by the plaintext stream as well. RC4 is an example of a well-known stream cipher.

  • Cryptographic hash functions (often called message digest functions) do not use keys, but are a related and important class of cryptographic algorithms. They take input data (often an entire message), and output a short, fixed length hash, and do so as a one-way function. For good ones, collisions (two plaintexts which produce the same hash) are extremely difficult to find.

  • Message authentication codes (MACs) are much like cryptographic hash functions, except that a secret key is used to authenticate the hash value on receipt.

Public-Key Cryptography

Symmetric-key cryptosystems typically use the same key for encryption and decryption, though this message or group of messages may have a different key than others. A significant disadvantage of symmetric ciphers is the key management necessary to use them securely. Each distinct pair of communicating parties must, ideally, share a different key, and perhaps each ciphertext exchanged as well. The number of keys required increases as the square of the number of network members, which very quickly requires complex key management schemes to keep them all straight and secret. The difficulty of establishing a secret key between two communicating parties, when a secure channel doesn't already exist between them, also presents a chicken-and-egg problem which is a considerable practical obstacle for cryptography users in the real world.

In a groundbreaking 1976 paper, Whitfield Diffie and Martin Hellman proposed the notion of public-key (also, more generally, called asymmetric key) cryptography in which two different but mathematically related keys are used - a public key and a private key. A public key system is so constructed that calculation of one key (the 'private key') is computationally infeasible from the other (the 'public key'), even though they are necessarily related. Instead, both keys are generated secretly, as an interrelated pair.

In public-key cryptosystems, the public key may be freely distributed, while its paired private key must remain secret. The public key is typically used for encryption, while the private or secret key is used for decryption. Diffie and Hellman showed that public-key cryptography was possible by presenting the Diffie-Hellman key exchange protocol.

The Diffie-Hellman and RSA algorithms, in addition to being the first publicly known examples of high quality public-key ciphers, have been among the most widely used. Others include the Cramer-Shoup cryptosystem, ElGamal encryption, and various elliptic curve techniques.

In addition to encryption, public-key cryptography can be used to implement digital signature schemes. A digital signature is reminiscent of an ordinary signature; they both have the characteristic that they are easy for a user to produce, but difficult for anyone else to forge. Digital signatures can also be permanently tied to the content of the message being signed; they cannot be 'moved' from one document to another, for any attempt will be detectable. In digital signature schemes, there are two algorithms: one for signing, in which a secret key is used to process the message (or a hash of the message, or both), and one for verification, in which the matching public key is used with the message to check the validity of the signature. RSA and Digital Signature Algorithm are two of the most popular digital signature schemes. Digital signatures are central to the operation of public key infrastructures and many network security schemes (SSL/TLS, many VPNs, etc).

Public-key algorithms are most often based on the computational complexity of "hard" problems, often from number theory. For example, the hardness of RSA is related to the integer factorization problem, while Diffie-Hellman and DSA are related to the discrete logarithm problem. More recently, elliptic curve cryptography has developed in which security is based on number theoretic problems involving elliptic curves. Because of the difficulty of the underlying problems, most public-key algorithms involve operations such as modular multiplication and exponentiation, which are much more computationally expensive than the techniques used in most block ciphers, especially with typical key sizes. As a result, public-key cryptosystems are commonly hybrid cryptosystems, in which a fast high-quality symmetric-key encryption algorithm is used for the message itself, while the relevant symmetric key is sent with the message, but encrypted using a public-key algorithm. Similarly, hybrid signature schemes are often used, in which a cryptographic hash function is computed, and only the resulting hash is digitally signed.


The goal of cryptanalysis is to find some weakness or insecurity in a cryptographic scheme, thus permitting its subversion or evasion. Cryptanalysis might be undertaken by a malicious attacker, attempting to subvert a system, or by the system's designer (or others) attempting to evaluate whether a system has vulnerabilities, and so it is not inherently a hostile act. In modern practice, however, cryptographic algorithms and protocols must be carefully examined and tested to offer any assurance of the system's security (at least, under clear - and hopefully reasonable - assumptions).

It is a commonly held misconception that every encryption method can be broken. In connection with his WWII work at Bell Labs, Claude Shannon proved that the one-time pad cipher is unbreakable, provided the key material is truly random, never reused, kept secret from all possible attackers, and of equal or greater length than the message. Most ciphers, apart from the one-time pad, can be broken with enough computational effort by brute force attack, but the amount of effort needed may be exponentially dependent on the key size, as compared to the effort needed to use the cipher. In such cases, effective security could be achieved if it is proven that the effort required (ie, 'work factor' in Shannon's terms) is beyond the ability of any adversary. This means it must be shown that no efficient method (as opposed to the time-consuming brute force method) can be found to break the cipher. Since no such showing can be made currently, as of today, the one-time-pad remains the only theoretically unbreakable cipher.

There are a wide variety of cryptanalytic attacks, and they can be classified in any of several ways. A common distinction turns on what an attacker knows and what capabilities are available. In a ciphertext-only attack, the cryptanalyst has access only to the ciphertext (good modern cryptosystems are usually effectively immune to ciphertext-only attacks). In a known-plaintext attack, the cryptanalyst has access to a ciphertext and its corresponding plaintext (or to many such pairs). In a chosen-plaintext attack, the cryptanalyst may choose a plaintext and learn its corresponding ciphertext (perhaps many times). Finally, in a chosen-ciphertext attack, the cryptanalyst may choose ciphertexts and learn their corresponding plaintexts. Also important, often overwhelmingly so, are mistakes (generally in the design or use of one of the protocols involved.

Cryptanalysis of symmetric-key ciphers typically involves looking for attacks against the block ciphers or stream ciphers that are more efficient than any attack that could be against a perfect cipher. For example, a simple brute force attack against DES requires one known plaintext and 255 decryptions, trying approximately half of the possible keys, to reach a point at which chances are better than even the key sought will have been found. But this may not be enough assurance; a linear cryptanalysis attack against DES requires 243 known plaintexts and approximately 243 DES operations. This is a considerable improvement on brute force attacks.

Public-key algorithms are based on the computational difficulty of various problems. The most famous of these is integer factorization (eg, the RSA algorithm is based on a problem related to factoring), but the discrete logarithm problem is also important. Much public-key cryptanalysis concerns numerical algorithms for solving these computational problems, or some of them, efficiently. For instance, the best known algorithms for solving the elliptic curve-based version of discrete logarithm are much more time-consuming than the best known algorithms for factoring, at least for problems of more or less equivalent size. Thus, other things being equal, to achieve an equivalent strength of attack resistance, factoring-based encryption techniques must use larger keys than elliptic curve techniques. For this reason, public-key cryptosystems based on elliptic curves have become popular since their invention in the mid-1990s.

While pure cryptanalysis uses weaknesses in the algorithms themselves, other attacks on cryptosystems are based on actual use of the algorithms in real devices, and are called side-channel attacks. If a cryptanalyst has access to, say, the amount of time the device took to encrypt a number of plaintexts or report an error in a password or PIN character, he may be able to use a timing attack to break a cipher that is otherwise resistant to analysis. An attacker might also study the pattern and length of messages to derive valuable information; this is known as traffic analysis, and can be quite useful to an alert adversary. And, of course, social engineering, and other attacks against the personnel who work with cryptosystems or the messages they handle (e.g., bribery, extortion, blackmail, espionage, ...) may be the most productive attacks of all.

Cryptographic Primitives

One-way functions are mathematical functions that are easy to compute but hard to invert. Much of the theoretical work in cryptography concerns cryptographic primitives- algorithms with basic cryptographic properties - and their relationship to other cryptographic problems. For example, a one-way function is a mathematical function intended to be easy to compute but hard to invert. In a very general sense, for any cryptographic application to be secure (if based on such computational feasibility assumptions), one-way functions must exist. However, if one-way functions exist, this implies that Complexity classes (P and NP|P ≠ NP). Since the P versus NP problem is currently unsolved, we don't know if one-way functions really do exist. For instance, if one-way functions exist, then secure pseudorandom generators and secure pseudorandom functions exist.

Currently known cryptographic primitives provide only basic functionality. These are usually noted as confidentiality, message integrity, authentication, and non-repudiation. Any other functionality in a cryptosystem must be built in using combinations of these algorithms and assorted protocols. Such combinations are called cryptosystems and it is they which users will encounter. Examples include PGP and its variants, SSH, SSL/TLS, all PKIs, digital signatures, etc. Other cryptographic primitives include the encryption algorithms themselves, one-way permutations, trapdoor permutations, etc.

Cryptographic Protocols

In many cases, cryptographic techniques involve back and forth communication among two or more parties in space (eg, between the home office and a branch office) or across time (e.g., cryptographically protected backup data). The term cryptographic protocol captures this general idea.

When the security of a good cryptographic system fails, it is rare that the vulnerability leading to the breach will have been in a quality cryptographic primitive. Instead, weaknesses are often mistakes in the protocol design (often due to inadequate design procedures, or less than thoroughly informed designers), in the implementation (e.g., a software bug), in a failure of the assumptions on which the design was based (e.g., proper training of those who will be using the system), or some other human error. Many cryptographic protocols have been designed and analyzed using ad hoc methods, but they rarely have any proof of security. Methods for formally analyzing the security of protocols, based on techniques from mathematical logic (see for example BAN logic), and more recently from concrete security principles, have been the subject of research for the past few decades. Unfortunately, to date these tools have been cumbersome and are not widely used for complex designs.

The study of how best to implement and integrate cryptography in applications is itself a distinct field, see: cryptographic engineering and security engineering.

Legal Issues Involving Cryptography


Cryptography has long been of interest to intelligence gathering agencies and law enforcement agencies. Because of its facilitation of privacy, and the diminution of privacy attendant on its prohibition, cryptography is also of considerable interest to civil rights supporters. Accordingly, there has been a history of controversial legal issues surrounding cryptography, especially since the advent of inexpensive computers has made possible widespread access to high quality cryptography.

In some countries, even the domestic use of cryptography is, or has been, restricted. Until 1999, France significantly restricted the use of cryptography domestically. In People's Republic of China, a license is still required to use cryptography. Many countries have tight restrictions on the use of cryptography. Among the more restrictive are laws in Belarus, Kazakhstan, Mongolia, Pakistan, Russia, Singapore, Tunisia, Venezuela, and Vietnam.

In the United States, cryptography is legal for domestic use, but there has been much conflict over legal issues related to cryptography. One particularly important issue has been the export of cryptography and cryptographic software and hardware. Because of the importance of cryptanalysis in World War II and an expectation that cryptography would continue to be important for national security, many western governments have, at some point, strictly regulated export of cryptography. After World War II, it was illegal in the US to sell or distribute encryption technology overseas; in fact, encryption was classified as a munition, like tanks and nuclear weapons Until the advent of the personal computer and the Internet, this was not especially problematic. Good cryptography is indistinguishable from bad cryptography for nearly all users, and in any case, most of the cryptographic techniques generally available were slow and error prone whether good or bad. However, as the Internet grew and computers became more widely available, high quality encryption techniques became well-known around the globe. As a result, export controls came to be seen to be an impediment to commerce and to research.

Export Controls

In 1996, thirty-nine countries signed the Wassenaar Arrangement, an arms control treaty that deals with the export of arms and "dual-use" technologies such as cryptography. The treaty stipulated that the use of cryptography with short key-lengths (56-bit for symmetric encryption, 512-bit for RSA) would no longer be export-controlled. The Wassenaar Arrangement on Export Controls for Conventional Arms and Dual-Use Goods and Technologies Cryptography exports from the US are now much less strictly regulated than in the past as a consequence of a major relaxation in 2000; there are no longer very many restrictions on key sizes in US exported mass market software. In practice today, since the relaxation in US export restrictions, and because almost every personal computer connected to the Internet, everywhere in the world, includes US sourced web browsers such as Mozilla Firefox or Microsoft Internet Explorer, almost every Internet user worldwide has access to quality cryptography (eg, using long keys at a minimum) in their browser's Transport Layer Security or SSL stack. The Mozilla Thunderbird and Microsoft Outlook E-mail client programs similarly can connect to IMAP or Post Office Protocol servers via TLS, and can send and receive email encrypted with S/MIME. Many Internet users don't realize that their basic application software contains such extensive cryptosystems. These browsers and email programs are so ubiquitous that even governments whose intent is to regulate civilian use of cryptography generally don't find it practical to do much to control distribution or use of cryptography of this quality, so even when such laws are in force, actual enforcement is often effectively impossible.

NSA Involvement

Another contentious issue connected to cryptography in the United States is the influence of the National Security Agency in cipher development and policy. NSA was involved with the design of Data Encryption Standard during its development at IBM and its consideration by the National Bureau of Standards as a possible Federal Standard for cryptography. DES was designed to be secure against differential cryptanalysis, a powerful and general cryptanalytic technique known to NSA and IBM, that became publicly known only when it was rediscovered in the late 1980s. IBM rediscovered differential cryptanalysis but kept the technique secret at NSA's request. The entire affair illustrates the difficulty of determining what resources and knowledge an attacker might actually have.

Another instance of NSA's involvement was the 1993 Clipper chip affair, an encryption microchip intended to be part of the Capstone cryptography-control initiative. Clipper was widely criticized by cryptographers for two reasons: the cipher algorithm was classified (the cipher, called Skipjack, was declassified in 1998 long after the Clipper initiative lapsed), which caused concerns that NSA had deliberately made the cipher weak in order to assist its intelligence efforts. The whole initiative was also criticized based on its violation of Kerckhoffs' principle, as the scheme included a special escrow key held by the government for use by law enforcement, for example in wiretaps.

Digital Rights Management

Cryptography is central to digital rights management (DRM), a group of techniques for technologically controlling use of copyrighted material, being widely implemented and deployed at the behest of some copyright holders. In 1998, President Bill Clinton signed the Digital Millennium Copyright Act (DMCA), which criminalized all production, dissemination, and use of certain cryptanalytic techniques and technology (now known or later discovered); specifically, those that could be used to circumvent DRM technological schemes. Digital Millennium Copyright Act This had a very serious potential impact on the cryptography research community since an argument can be made that any cryptanalytic research violated, or might violate, the DMCA. The Federal Bureau of Investigation and the Justice Department have not enforced the DMCA as rigorously as had been feared by some, but the law, nonetheless, remains a controversial one. One well-respected cryptography researcher, Niels Ferguson, has publicly stated that he will not release some research into an Intel Corporation security design for fear of prosecution under the DMCA, and both Alan Cox (longtime number two in Linux kernel development) and Professor Edward Felten (and some of his students at Princeton) have encountered problems related to the Act. Dmitry Sklyarov was arrested during a visit to the US from Russia, and jailed for some months for alleged violations of the DMCA which had occurred in Russia, where the work for which he was arrested and charged was then, and when he was arrested, legal. Similar statutes have since been enacted in several countries. See for instance the Directive on the harmonization of certain aspects of copyright and related rights in the information society. In 2007, the cryptographic keys responsible for DVD and HDDVD content scrambling were discovered and released onto the Internet. Both times, the MPAA sent out numerous DMCA take-down notices, and there was a AACS encryption key controversy and also a massive Internet backlash as a result of the implications of such notices on fair use and free speech.

See Also

  • International Association for Cryptologic Research.
  • Chaos Computer Club.

Further Reading

  • The Codebreakers by David Kahn, a comprehensive history of classical (pre-WW2) cryptography. The current edition has a brief addendum about WW2 and later.
  • The Code Book by Simon Singh, a clearly written anecdotal history of crypto, covering modern methods including public key.
  • Crypto: How the Code Rebels Beat the Government Saving Privacy in the Digital Age]] by Steven Levy, about the political and legal conflicts in the US about cryptography, such as the Clipper Chip controversy and the Bernstein v. United States lawsuit.
  • Applied Cryptography, 2nd edition, by Bruce Schneier. General reference book about crypto algorithms and protocols, aimed at implementers.
  • Handbook of Applied Cryptography by A. J. Menezes, P. C. van Oorschot, and S. A. Vanstone CRC Press, (PDF download available), somewhat more mathematical than Schneier's Applied Cryptography.
  • Introduction to Modern Cryptography by Phillip Rogaway and Mihir Bellare, a mathematical introduction to theoretical cryptography including reduction-based security proofs. PDF download.
  • Stealing Secrets, Telling Lies: How Spies and Codebreakers Helped Shape the Twentieth Century, by James Gannon.
  • Alvin's Secret Code by Clifford B. Hicks (children's novel that introduces some basic cryptography and cryptanalysis).
  • In Code: A Mathematical Journey by Sarah Flannery (with David Flannery). Popular account of Sarah's award-winning project on public-key cryptography, co-written with her father.
  • Cryptography and Mathematics by Bernhard Esslinger, 200 pages, part of the free open-source package Cryptool,
  • Ibrahim A. Al-Kadi ,"The origins of cryptology: The Arab contributions”, Cryptologia, 16(2) (April 1992) pp. 97–126.
  • Andreas Pfitzmann: Security in IT Networks: Multilateral Security in Distributed and by Distributed Systems

External References