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Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with
cryptographic Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...
algorithms In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
based on
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
problems termed one-way functions. Security of public-key cryptography depends on keeping the private key secret; the public key can be openly distributed without compromising security. In a public-key encryption system, anyone with a public key can encrypt a message, yielding a ciphertext, but only those who know the corresponding private key can decrypt the ciphertext to obtain the original message. For example, a journalist can publish the public key of an encryption key pair on a web site so that sources can send secret messages to the news organization in ciphertext. Only the journalist who knows the corresponding private key can decrypt the ciphertexts to obtain the sources' messages—an eavesdropper reading email on its way to the journalist cannot decrypt the ciphertexts. However, public-key encryption does not conceal metadata like what computer a source used to send a message, when they sent it, or how long it is. Public-key encryption on its own also does not tell the recipient anything about who sent a message—it just conceals the content of a message in a ciphertext that can only be decrypted with the private key. In a
digital signature A digital signature is a mathematical scheme for verifying the authenticity of digital messages or documents. A valid digital signature, where the prerequisites are satisfied, gives a recipient very high confidence that the message was created b ...
system, a sender can use a private key together with a message to create a signature. Anyone with the corresponding public key can verify whether the signature matches the message, but a forger who does not know the private key cannot find any message/signature pair that will pass verification with the public key. For example, a software publisher can create a signature key pair and include the public key in software installed on computers. Later, the publisher can distribute an update to the software signed using the private key, and any computer receiving an update can confirm it is genuine by verifying the signature using the public key. As long as the software publisher keeps the private key secret, even if a forger can distribute malicious updates to computers, they cannot convince the computers that any malicious updates are genuine. Public key algorithms are fundamental security primitives in modern
cryptosystem In cryptography, a cryptosystem is a suite of cryptographic algorithms needed to implement a particular security service, such as confidentiality (encryption). Typically, a cryptosystem consists of three algorithms: one for key generation, one f ...
s, including applications and protocols which offer assurance of the confidentiality, authenticity and non-repudiability of electronic communications and data storage. They underpin numerous Internet standards, such as Transport Layer Security (TLS), SSH,
S/MIME S/MIME (Secure/Multipurpose Internet Mail Extensions) is a standard for public key encryption and signing of MIME data. S/MIME is on an IETF standards track and defined in a number of documents, most importantly . It was originally developed b ...
and PGP. Some public key algorithms provide key distribution and secrecy (e.g.,
Diffie–Hellman key exchange Diffie–Hellman key exchangeSynonyms of Diffie–Hellman key exchange include: * Diffie–Hellman–Merkle key exchange * Diffie–Hellman key agreement * Diffie–Hellman key establishment * Diffie–Hellman key negotiation * Exponential key exc ...
), some provide
digital signature A digital signature is a mathematical scheme for verifying the authenticity of digital messages or documents. A valid digital signature, where the prerequisites are satisfied, gives a recipient very high confidence that the message was created b ...
s (e.g.,
Digital Signature Algorithm The Digital Signature Algorithm (DSA) is a public-key cryptosystem and Federal Information Processing Standard for digital signatures, based on the mathematical concept of modular exponentiation and the discrete logarithm problem. DSA is a varia ...
), and some provide both (e.g.,
RSA RSA may refer to: Organizations Academia and education * Rabbinical Seminary of America, a yeshiva in New York City *Regional Science Association International (formerly the Regional Science Association), a US-based learned society *Renaissance S ...
). Compared to symmetric encryption, asymmetric encryption is rather slower than good symmetric encryption, too slow for many purposes. Today's cryptosystems (such as
TLS TLS may refer to: Computing * Transport Layer Security, a cryptographic protocol for secure computer network communication * Thread level speculation, an optimisation on multiprocessor CPUs * Thread-local storage, a mechanism for allocating vari ...
,
Secure Shell The Secure Shell Protocol (SSH) is a cryptographic network protocol for operating network services securely over an unsecured network. Its most notable applications are remote login and command-line execution. SSH applications are based ...
) use both symmetric encryption and asymmetric encryption, often by using asymmetric encryption to securely exchange a secret key which is then used for symmetric encryption.


Description

Before the mid-1970s, all cipher systems used symmetric key algorithms, in which the same cryptographic key is used with the underlying algorithm by both the sender and the recipient, who must both keep it secret. Of necessity, the key in every such system had to be exchanged between the communicating parties in some secure way prior to any use of the system – for instance, via a secure channel. This requirement is never trivial and very rapidly becomes unmanageable as the number of participants increases, or when secure channels are not available, or when, (as is sensible cryptographic practice), keys are frequently changed. In particular, if messages are meant to be secure from other users, a separate key is required for each possible pair of users. By contrast, in a public key system, the public keys can be disseminated widely and openly, and only the corresponding private keys need be kept secret by its owner. Two of the best-known uses of public key cryptography are: * ''Public key encryption'', in which a message is encrypted with the intended recipient's public key. For properly chosen and used algorithms, messages cannot in practice be decrypted by anyone who does not possess the matching private key, who is thus presumed to be the owner of that key and so the person associated with the public key. This can be used to ensure
confidentiality Confidentiality involves a set of rules or a promise usually executed through confidentiality agreements that limits the access or places restrictions on certain types of information. Legal confidentiality By law, lawyers are often required ...
of a message. * ''
Digital signature A digital signature is a mathematical scheme for verifying the authenticity of digital messages or documents. A valid digital signature, where the prerequisites are satisfied, gives a recipient very high confidence that the message was created b ...
s'', in which a message is signed with the sender's private key and can be verified by anyone who has access to the sender's public key. This verification proves that the sender had access to the private key, and therefore is very likely to be the person associated with the public key. It also proves that the signature was prepared for that exact message, since verification will fail for any other message one could devise without using the private key. One important issue is confidence/proof that a particular public key is authentic, i.e. that it is correct and belongs to the person or entity claimed, and has not been tampered with or replaced by some (perhaps malicious) third party. There are several possible approaches, including: A
public key infrastructure A public key infrastructure (PKI) is a set of roles, policies, hardware, software and procedures needed to create, manage, distribute, use, store and revoke digital certificates and manage public-key encryption. The purpose of a PKI is to facil ...
(PKI), in which one or more third parties – known as
certificate authorities In cryptography, a certificate authority or certification authority (CA) is an entity that stores, signs, and issues digital certificates. A digital certificate certifies the ownership of a public key by the named subject of the certificate. This ...
– certify ownership of key pairs.
TLS TLS may refer to: Computing * Transport Layer Security, a cryptographic protocol for secure computer network communication * Thread level speculation, an optimisation on multiprocessor CPUs * Thread-local storage, a mechanism for allocating vari ...
relies upon this. This implies that the PKI system (software, hardware, and management) is trust-able by all involved. A "
web of trust In cryptography, a web of trust is a concept used in PGP, GnuPG, and other OpenPGP-compatible systems to establish the authenticity of the binding between a public key and its owner. Its decentralized trust model is an alternative to the ...
" which decentralizes authentication by using individual endorsements of links between a user and the public key belonging to that user. PGP uses this approach, in addition to lookup in the
domain name system The Domain Name System (DNS) is a hierarchical and distributed naming system for computers, services, and other resources in the Internet or other Internet Protocol (IP) networks. It associates various information with domain names assigned ...
(DNS). The DKIM system for digitally signing emails also uses this approach.


Applications

The most obvious application of a public key encryption system is for encrypting communication to provide
confidentiality Confidentiality involves a set of rules or a promise usually executed through confidentiality agreements that limits the access or places restrictions on certain types of information. Legal confidentiality By law, lawyers are often required ...
– a message that a sender encrypts using the recipient's public key which can be decrypted only by the recipient's paired private key. Another application in public key cryptography is the
digital signature A digital signature is a mathematical scheme for verifying the authenticity of digital messages or documents. A valid digital signature, where the prerequisites are satisfied, gives a recipient very high confidence that the message was created b ...
. Digital signature schemes can be used for sender
authentication Authentication (from ''authentikos'', "real, genuine", from αὐθέντης ''authentes'', "author") is the act of proving an assertion, such as the identity of a computer system user. In contrast with identification, the act of indicat ...
. Non-repudiation systems use digital signatures to ensure that one party cannot successfully dispute its authorship of a document or communication. Further applications built on this foundation include: digital cash, password-authenticated key agreement, time-stamping services and non-repudiation protocols.


Hybrid cryptosystems

Because asymmetric key algorithms are nearly always much more computationally intensive than symmetric ones, it is common to use a public/private ''asymmetric''
key-exchange algorithm Key exchange (also key establishment) is a method in cryptography by which cryptographic keys are exchanged between two parties, allowing use of a cryptographic algorithm. If the sender and receiver wish to exchange encrypted messages, each m ...
to encrypt and exchange a ''symmetric key'', which is then used by
symmetric-key cryptography Symmetric-key algorithms are algorithms for cryptography that use the same cryptographic keys for both the encryption of plaintext and the decryption of ciphertext. The keys may be identical, or there may be a simple transformation to go between ...
to transmit data using the now-shared ''symmetric key'' for a symmetric key encryption algorithm. PGP, SSH, and the SSL/TLS family of schemes use this procedure; they are thus called ''
hybrid cryptosystem In cryptography, a hybrid cryptosystem is one which combines the convenience of a public-key cryptosystem with the efficiency of a symmetric-key cryptosystem. Public-key cryptosystems are convenient in that they do not require the sender and receiv ...
s''. The initial ''asymmetric'' cryptography-based key exchange to share a server-generated ''symmetric'' key from the server to client has the advantage of not requiring that a symmetric key be pre-shared manually, such as on printed paper or discs transported by a courier, while providing the higher data throughput of symmetric key cryptography over asymmetric key cryptography for the remainder of the shared connection.


Weaknesses

As with all security-related systems, it is important to identify potential weaknesses. Aside from poor choice of an asymmetric key algorithm (there are few which are widely regarded as satisfactory) or too short a key length, the chief security risk is that the private key of a pair becomes known. All security of messages, authentication, etc., will then be lost.


Algorithms

All public key schemes are in theory susceptible to a " brute-force key search attack". However, such an attack is impractical if the amount of computation needed to succeed – termed the "work factor" by
Claude Shannon Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician, electrical engineer, and cryptographer known as a "father of information theory". As a 21-year-old master's degree student at the Massachusetts In ...
– is out of reach of all potential attackers. In many cases, the work factor can be increased by simply choosing a longer key. But other algorithms may inherently have much lower work factors, making resistance to a brute-force attack (e.g., from longer keys) irrelevant. Some special and specific algorithms have been developed to aid in attacking some public key encryption algorithms; both
RSA RSA may refer to: Organizations Academia and education * Rabbinical Seminary of America, a yeshiva in New York City *Regional Science Association International (formerly the Regional Science Association), a US-based learned society *Renaissance S ...
and ElGamal encryption have known attacks that are much faster than the brute-force approach. None of these are sufficiently improved to be actually practical, however. Major weaknesses have been found for several formerly promising asymmetric key algorithms. The "knapsack packing" algorithm was found to be insecure after the development of a new attack. As with all cryptographic functions, public-key implementations may be vulnerable to side-channel attacks that exploit information leakage to simplify the search for a secret key. These are often independent of the algorithm being used. Research is underway to both discover, and to protect against, new attacks.


Alteration of public keys

Another potential security vulnerability in using asymmetric keys is the possibility of a "man-in-the-middle" attack, in which the communication of public keys is intercepted by a third party (the "man in the middle") and then modified to provide different public keys instead. Encrypted messages and responses must, in all instances, be intercepted, decrypted, and re-encrypted by the attacker using the correct public keys for the different communication segments so as to avoid suspicion. A communication is said to be insecure where data is transmitted in a manner that allows for interception (also called " sniffing"). These terms refer to reading the sender's private data in its entirety. A communication is particularly unsafe when interceptions can't be prevented or monitored by the sender. A man-in-the-middle attack can be difficult to implement due to the complexities of modern security protocols. However, the task becomes simpler when a sender is using insecure media such as public networks, the
Internet The Internet (or internet) is the global system of interconnected computer networks that uses the Internet protocol suite (TCP/IP) to communicate between networks and devices. It is a ''internetworking, network of networks'' that consists ...
, or wireless communication. In these cases an attacker can compromise the communications infrastructure rather than the data itself. A hypothetical malicious staff member at an
Internet Service Provider An Internet service provider (ISP) is an organization that provides services for accessing, using, or participating in the Internet. ISPs can be organized in various forms, such as commercial, community-owned, non-profit, or otherwise privatel ...
(ISP) might find a man-in-the-middle attack relatively straightforward. Capturing the public key would only require searching for the key as it gets sent through the ISP's communications hardware; in properly implemented asymmetric key schemes, this is not a significant risk. In some advanced man-in-the-middle attacks, one side of the communication will see the original data while the other will receive a malicious variant. Asymmetric man-in-the-middle attacks can prevent users from realizing their connection is compromised. This remains so even when one user's data is known to be compromised because the data appears fine to the other user. This can lead to confusing disagreements between users such as "it must be on your end!" when neither user is at fault. Hence, man-in-the-middle attacks are only fully preventable when the communications infrastructure is physically controlled by one or both parties; such as via a wired route inside the sender's own building. In summation, public keys are easier to alter when the communications hardware used by a sender is controlled by an attacker.


Public key infrastructure

One approach to prevent such attacks involves the use of a
public key infrastructure A public key infrastructure (PKI) is a set of roles, policies, hardware, software and procedures needed to create, manage, distribute, use, store and revoke digital certificates and manage public-key encryption. The purpose of a PKI is to facil ...
(PKI); a set of roles, policies, and procedures needed to create, manage, distribute, use, store and revoke digital certificates and manage public-key encryption. However, this has potential weaknesses. For example, the certificate authority issuing the certificate must be trusted by all participating parties to have properly checked the identity of the key-holder, to have ensured the correctness of the public key when it issues a certificate, to be secure from computer piracy, and to have made arrangements with all participants to check all their certificates before protected communications can begin.
Web browser A web browser is application software for accessing websites. When a user requests a web page from a particular website, the browser retrieves its files from a web server and then displays the page on the user's screen. Browsers are used on ...
s, for instance, are supplied with a long list of "self-signed identity certificates" from PKI providers – these are used to check the ''bona fides'' of the certificate authority and then, in a second step, the certificates of potential communicators. An attacker who could subvert one of those certificate authorities into issuing a certificate for a bogus public key could then mount a "man-in-the-middle" attack as easily as if the certificate scheme were not used at all. In an alternative scenario rarely discussed, an attacker who penetrates an authority's servers and obtains its store of certificates and keys (public and private) would be able to spoof, masquerade, decrypt, and forge transactions without limit. Despite its theoretical and potential problems, this approach is widely used. Examples include
TLS TLS may refer to: Computing * Transport Layer Security, a cryptographic protocol for secure computer network communication * Thread level speculation, an optimisation on multiprocessor CPUs * Thread-local storage, a mechanism for allocating vari ...
and its predecessor
SSL SSL may refer to: Entertainment * RoboCup Small Size League, robotics football competition * ''Sesame Street Live'', a touring version of the children's television show * StarCraft II StarLeague, a Korean league in the video game Natural language ...
, which are commonly used to provide security for web browser transactions (for example, to securely send credit card details to an online store). Aside from the resistance to attack of a particular key pair, the security of the certification
hierarchy A hierarchy (from Greek: , from , 'president of sacred rites') is an arrangement of items (objects, names, values, categories, etc.) that are represented as being "above", "below", or "at the same level as" one another. Hierarchy is an important ...
must be considered when deploying public key systems. Some certificate authority – usually a purpose-built program running on a server computer – vouches for the identities assigned to specific private keys by producing a digital certificate. Public key digital certificates are typically valid for several years at a time, so the associated private keys must be held securely over that time. When a private key used for certificate creation higher in the PKI server hierarchy is compromised, or accidentally disclosed, then a "
man-in-the-middle attack In cryptography and computer security, a man-in-the-middle, monster-in-the-middle, machine-in-the-middle, monkey-in-the-middle, meddler-in-the-middle, manipulator-in-the-middle (MITM), person-in-the-middle (PITM) or adversary-in-the-middle (AiTM) ...
" is possible, making any subordinate certificate wholly insecure.


Examples

Examples of well-regarded asymmetric key techniques for varied purposes include: *
Diffie–Hellman key exchange Diffie–Hellman key exchangeSynonyms of Diffie–Hellman key exchange include: * Diffie–Hellman–Merkle key exchange * Diffie–Hellman key agreement * Diffie–Hellman key establishment * Diffie–Hellman key negotiation * Exponential key exc ...
protocol *DSS (Digital Signature Standard), which incorporates the
Digital Signature Algorithm The Digital Signature Algorithm (DSA) is a public-key cryptosystem and Federal Information Processing Standard for digital signatures, based on the mathematical concept of modular exponentiation and the discrete logarithm problem. DSA is a varia ...
*
ElGamal In cryptography, the ElGamal encryption system is an asymmetric key encryption algorithm for public-key cryptography which is based on the Diffie–Hellman key exchange. It was described by Taher Elgamal in 1985. ElGamal encryption is used in t ...
*
Elliptic-curve cryptography Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys compared to non-EC cryptography (based on plain Galois fields) to provide e ...
**
Elliptic Curve Digital Signature Algorithm In cryptography, the Elliptic Curve Digital Signature Algorithm (ECDSA) offers a variant of the Digital Signature Algorithm (DSA) which uses elliptic-curve cryptography. Key and signature-size As with elliptic-curve cryptography in general, the ...
(ECDSA) ** Elliptic-curve Diffie–Hellman (ECDH) ** Ed25519 and Ed448 ( EdDSA) ** X25519 and X448 (ECDH/EdDH) *Various password-authenticated key agreement techniques * Paillier cryptosystem *
RSA RSA may refer to: Organizations Academia and education * Rabbinical Seminary of America, a yeshiva in New York City *Regional Science Association International (formerly the Regional Science Association), a US-based learned society *Renaissance S ...
encryption algorithm ( PKCS#1) *
Cramer–Shoup cryptosystem The Cramer–Shoup system is an asymmetric key encryption algorithm, and was the first efficient scheme proven to be secure against adaptive chosen ciphertext attack using standard cryptographic assumptions. Its security is based on the computation ...
* YAK authenticated key agreement protocol Examples of asymmetric key algorithms not yet widely adopted include: * NTRUEncrypt cryptosystem * Kyber *
McEliece cryptosystem In cryptography, the McEliece cryptosystem is an asymmetric encryption algorithm developed in 1978 by Robert McEliece. It was the first such scheme to use randomization in the encryption process. The algorithm has never gained much acceptance in ...
Examples of notable – yet insecure – asymmetric key algorithms include: * Merkle–Hellman knapsack cryptosystem Examples of protocols using asymmetric key algorithms include: *
S/MIME S/MIME (Secure/Multipurpose Internet Mail Extensions) is a standard for public key encryption and signing of MIME data. S/MIME is on an IETF standards track and defined in a number of documents, most importantly . It was originally developed b ...
* GPG, an implementation of OpenPGP, and an Internet Standard * EMV, EMV Certificate Authority *
IPsec In computing, Internet Protocol Security (IPsec) is a secure network protocol suite that authenticates and encrypts packets of data to provide secure encrypted communication between two computers over an Internet Protocol network. It is used in ...
* PGP * ZRTP, a secure
VoIP Voice over Internet Protocol (VoIP), also called IP telephony, is a method and group of technologies for the delivery of voice communications and multimedia sessions over Internet Protocol (IP) networks, such as the Internet. The terms Interne ...
protocol *
Transport Layer Security Transport Layer Security (TLS) is a cryptographic protocol designed to provide communications security over a computer network. The protocol is widely used in applications such as email, instant messaging, and voice over IP, but its use in secu ...
standardized by
IETF The Internet Engineering Task Force (IETF) is a standards organization for the Internet and is responsible for the technical standards that make up the Internet protocol suite (TCP/IP). It has no formal membership roster or requirements and ...
and its predecessor Secure Socket Layer * SILC * SSH *
Bitcoin Bitcoin ( abbreviation: BTC; sign: ₿) is a decentralized digital currency that can be transferred on the peer-to-peer bitcoin network. Bitcoin transactions are verified by network nodes through cryptography and recorded in a public di ...
* Off-the-Record Messaging


History

During the early
history of cryptography Cryptography, the use of codes and ciphers to protect secrets, began thousands of years ago. Until recent decades, it has been the story of what might be called classical cryptography — that is, of methods of encryption that use pen and paper, ...
, two parties would rely upon a key that they would exchange by means of a secure, but non-cryptographic, method such as a face-to-face meeting, or a trusted courier. This key, which both parties must then keep absolutely secret, could then be used to exchange encrypted messages. A number of significant practical difficulties arise with this approach to distributing keys.


Anticipation

In his 1874 book ''The Principles of Science'',
William Stanley Jevons William Stanley Jevons (; 1 September 183513 August 1882) was an English economist and logician. Irving Fisher described Jevons's book ''A General Mathematical Theory of Political Economy'' (1862) as the start of the mathematical method in ec ...
wrote:
Can the reader say what two numbers multiplied together will produce the number 8616460799? I think it unlikely that anyone but myself will ever know.
Here he described the relationship of one-way functions to cryptography, and went on to discuss specifically the factorization problem used to create a trapdoor function. In July 1996, mathematician
Solomon W. Golomb Solomon Wolf Golomb (; May 30, 1932 – May 1, 2016) was an American mathematician, engineer, and professor of electrical engineering at the University of Southern California, best known for his works on mathematical games. Most notably, he inve ...
said: "Jevons anticipated a key feature of the RSA Algorithm for public key cryptography, although he certainly did not invent the concept of public key cryptography."


Classified discovery

In 1970,
James H. Ellis James Henry Ellis (25 September 1924 – 25 November 1997) was a British engineer and cryptographer. In 1970, while working at the Government Communications Headquarters (GCHQ) in Cheltenham, he conceived of the possibility of "non-secret encr ...
, a British cryptographer at the UK
Government Communications Headquarters Government Communications Headquarters, commonly known as GCHQ, is an intelligence and security organisation responsible for providing signals intelligence (SIGINT) and information assurance (IA) to the government and armed forces of the ...
(GCHQ), conceived of the possibility of "non-secret encryption", (now called public key cryptography), but could see no way to implement it. In 1973, his colleague Clifford Cocks implemented what has become known as the RSA encryption algorithm, giving a practical method of "non-secret encryption", and in 1974 another GCHQ mathematician and cryptographer,
Malcolm J. Williamson Malcolm John Williamson (2 November 1950 – 15 September 2015) was a British mathematician and cryptographer. In 1974 he developed what is now known as Diffie–Hellman key exchange. He was then working at Government Communications Headquarters ...
, developed what is now known as
Diffie–Hellman key exchange Diffie–Hellman key exchangeSynonyms of Diffie–Hellman key exchange include: * Diffie–Hellman–Merkle key exchange * Diffie–Hellman key agreement * Diffie–Hellman key establishment * Diffie–Hellman key negotiation * Exponential key exc ...
. The scheme was also passed to the US's
National Security Agency The National Security Agency (NSA) is a national-level intelligence agency of the United States Department of Defense, under the authority of the Director of National Intelligence (DNI). The NSA is responsible for global monitoring, collectio ...
. Both organisations had a military focus and only limited computing power was available in any case; the potential of public key cryptography remained unrealised by either organization:
I judged it most important for military use ... if you can share your key rapidly and electronically, you have a major advantage over your opponent. Only at the end of the evolution from Berners-Lee designing an open internet architecture for CERN, its adaptation and adoption for the
Arpanet The Advanced Research Projects Agency Network (ARPANET) was the first wide-area packet-switched network with distributed control and one of the first networks to implement the TCP/IP protocol suite. Both technologies became the technical foun ...
... did public key cryptography realise its full potential. — Ralph Benjamin
These discoveries were not publicly acknowledged for 27 years, until the research was declassified by the British government in 1997.


Public discovery

In 1976, an asymmetric key cryptosystem was published by Whitfield Diffie and
Martin Hellman Martin Edward Hellman (born October 2, 1945) is an American cryptologist and mathematician, best known for his involvement with public key cryptography in cooperation with Whitfield Diffie and Ralph Merkle. Hellman is a longtime contributor to ...
who, influenced by
Ralph Merkle Ralph C. Merkle (born February 2, 1952) is a computer scientist and mathematician. He is one of the inventors of public-key cryptography, the inventor of cryptographic hashing, and more recently a researcher and speaker on cryonics. Contribution ...
's work on public key distribution, disclosed a method of public key agreement. This method of key exchange, which uses exponentiation in a finite field, came to be known as
Diffie–Hellman key exchange Diffie–Hellman key exchangeSynonyms of Diffie–Hellman key exchange include: * Diffie–Hellman–Merkle key exchange * Diffie–Hellman key agreement * Diffie–Hellman key establishment * Diffie–Hellman key negotiation * Exponential key exc ...
. This was the first published practical method for establishing a shared secret-key over an authenticated (but not confidential) communications channel without using a prior shared secret. Merkle's "public key-agreement technique" became known as
Merkle's Puzzles In cryptography, Merkle's Puzzles is an early construction for a public-key cryptosystem, a protocol devised by Ralph Merkle in 1974 and published in 1978. It allows two parties to agree on a shared secret by exchanging messages, even if they ha ...
, and was invented in 1974 and only published in 1978. This makes asymmetric encryption a rather new field in cryptography although cryptography itself dates back more than 2,000 years. In 1977, a generalization of Cocks' scheme was independently invented by
Ron Rivest Ronald Linn Rivest (; born May 6, 1947) is a cryptographer and an Institute Professor at MIT. He is a member of MIT's Department of Electrical Engineering and Computer Science (EECS) and a member of MIT's Computer Science and Artificial Int ...
,
Adi Shamir Adi Shamir ( he, עדי שמיר; born July 6, 1952) is an Israeli cryptographer. He is a co-inventor of the Rivest–Shamir–Adleman (RSA) algorithm (along with Ron Rivest and Len Adleman), a co-inventor of the Feige–Fiat–Shamir identifica ...
and
Leonard Adleman Leonard Adleman (born December 31, 1945) is an American computer scientist. He is one of the creators of the RSA encryption algorithm, for which he received the 2002 Turing Award, often called the Nobel prize of Computer science. He is also know ...
, all then at MIT. The latter authors published their work in 1978 in
Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of L ...
's
Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it i ...
column, and the algorithm came to be known as
RSA RSA may refer to: Organizations Academia and education * Rabbinical Seminary of America, a yeshiva in New York City *Regional Science Association International (formerly the Regional Science Association), a US-based learned society *Renaissance S ...
, from their initials. RSA uses exponentiation modulo a product of two very large primes, to encrypt and decrypt, performing both public key encryption and public key digital signatures. Its security is connected to the extreme difficulty of factoring large integers, a problem for which there is no known efficient general technique (though prime factorization may be obtained through brute-force attacks; this grows much more difficult the larger the prime factors are). A description of the algorithm was published in the
Mathematical Games A mathematical game is a game whose rules, strategies, and outcomes are defined by clear mathematical parameters. Often, such games have simple rules and match procedures, such as Tic-tac-toe and Dots and Boxes. Generally, mathematical games ne ...
column in the August 1977 issue of
Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it i ...
. Since the 1970s, a large number and variety of encryption, digital signature, key agreement, and other techniques have been developed, including the Rabin cryptosystem, ElGamal encryption, DSA - and
elliptic curve cryptography Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys compared to non-EC cryptography (based on plain Galois fields) to provide e ...
.


See also

* Books on cryptography *
GNU Privacy Guard GNU Privacy Guard (GnuPG or GPG) is a free-software replacement for Symantec's PGP cryptographic software suite. The software is compliant with RFC 4880, the IETF standards-track specification of OpenPGP. Modern versions of PGP are interoper ...
*
Identity-based encryption ID-based encryption, or identity-based encryption (IBE), is an important primitive of ID-based cryptography. As such it is a type of public-key encryption in which the public key of a user is some unique information about the identity of the use ...
(IBE) * Key escrow * Key-agreement protocol *
PGP word list The PGP Word List (" Pretty Good Privacy word list", also called a biometric word list for reasons explained below) is a list of words for conveying data bytes in a clear unambiguous way via a voice channel. They are analogous in purpose to the N ...
* Post-quantum cryptography *
Pretty Good Privacy Pretty Good Privacy (PGP) is an encryption program that provides cryptographic privacy and authentication for data communication. PGP is used for signing, encrypting, and decrypting texts, e-mails, files, directories, and whole disk part ...
*
Pseudonym A pseudonym (; ) or alias () is a fictitious name that a person or group assumes for a particular purpose, which differs from their original or true name (orthonym). This also differs from a new name that entirely or legally replaces an individua ...
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Public key fingerprint In public-key cryptography, a public key fingerprint is a short sequence of bytes used to identify a longer public key. Fingerprints are created by applying a cryptographic hash function to a public key. Since fingerprints are shorter than the ...
*
Public key infrastructure A public key infrastructure (PKI) is a set of roles, policies, hardware, software and procedures needed to create, manage, distribute, use, store and revoke digital certificates and manage public-key encryption. The purpose of a PKI is to facil ...
(PKI) *
Quantum computing Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Thou ...
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Quantum cryptography Quantum cryptography is the science of exploiting quantum mechanical properties to perform cryptographic tasks. The best known example of quantum cryptography is quantum key distribution which offers an information-theoretically secure solutio ...
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Secure Shell The Secure Shell Protocol (SSH) is a cryptographic network protocol for operating network services securely over an unsecured network. Its most notable applications are remote login and command-line execution. SSH applications are based ...
(SSH) *
Symmetric-key algorithm Symmetric-key algorithms are algorithms for cryptography that use the same cryptographic keys for both the encryption of plaintext and the decryption of ciphertext. The keys may be identical, or there may be a simple transformation to go between ...
* Threshold cryptosystem *
Web of trust In cryptography, a web of trust is a concept used in PGP, GnuPG, and other OpenPGP-compatible systems to establish the authenticity of the binding between a public key and its owner. Its decentralized trust model is an alternative to the ...


Notes


References

*. The first two sections contain a very good introduction to public-key cryptography. * * *
IEEE 1363: Standard Specifications for Public-Key Cryptography
* Christof Paar, Jan Pelzl
"Introduction to Public-Key Cryptography"
Chapter 6 of "Understanding Cryptography, A Textbook for Students and Practitioners". (companion web site contains online cryptography course that covers public-key cryptography), Springer, 2009. *


External links


Oral history interview with Martin Hellman
Charles Babbage Institute The IT History Society (ITHS) is an organization that supports the history and scholarship of information technology by encouraging, fostering, and facilitating archival and historical research. Formerly known as the Charles Babbage Foundation, ...
, University of Minnesota. Leading cryptography scholar
Martin Hellman Martin Edward Hellman (born October 2, 1945) is an American cryptologist and mathematician, best known for his involvement with public key cryptography in cooperation with Whitfield Diffie and Ralph Merkle. Hellman is a longtime contributor to ...
discusses the circumstances and fundamental insights of his invention of public key cryptography with collaborators Whitfield Diffie and
Ralph Merkle Ralph C. Merkle (born February 2, 1952) is a computer scientist and mathematician. He is one of the inventors of public-key cryptography, the inventor of cryptographic hashing, and more recently a researcher and speaker on cryonics. Contribution ...
at Stanford University in the mid-1970s.
An account of how GCHQ kept their invention of PKE secret until 1997
{{Authority control Anonymity networks Cryptographic software Cryptographic protocols Cryptography Banking technology Public key infrastructure Network architecture