- Key People:
- Leonard M. Adleman
- Ronald L. Rivest
- Adi Shamir
- Related Topics:
- RSA encryption
- cipher
- On the Web:
- IOPscience - Journal of Physics: Conference Series - Public Key Cryptography (Oct. 25, 2026)
public-key cryptography
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What is cryptology?
Cryptology is the science of secure data communication and storage, encompassing cryptography and cryptanalysis. Cryptography focuses on concealing information using ciphers and keys, while cryptanalysis involves recovering secured information without knowing the key. Modern cryptology has expanded to include digital signatures, authentication, and other functions for the information age.
Public-key cryptography emerged as a significant advancement, but it is only one aspect of cryptology's evolution. Secret-sharing, a tool in modern cryptology, enhances security by requiring multiple participants to combine their information to decipher a key.
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What is public-key cryptography?
Public-key cryptography, or asymmetric cryptography, is a system where message transmitters and recipients use different keys. This eliminates the need to transmit codes, reducing the risk of interception. Each user keeps their decryption key private and publishes their encryption key in a public directory. Anyone can encrypt a message using the recipient's public key, but only the intended receiver can decrypt it.
The RSA cryptoalgorithm is a well-known public-key scheme where a user chooses two large prime numbers, (p) and (q), and publishes their product, (n = pq). The security of the algorithm relies on the difficulty of factoring (n). Public-key cryptography is often used in hybrid systems for authentication, digital signatures, or to exchange session keys for single-key algorithms. It is also commonly used in cryptocurrency transactions.
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How is public-key cryptography different from secret-key cryptography?
In secret-key (symmetric) cryptosystems, the transmitter and receiver use the same key, while in public-key (asymmetric) cryptosystems, they use different keys.
Secret-key cryptography suffers from key distribution problems because participants must initially exchange the same secret key. Public-key cryptography uses two keys, a public key for encryption and a private key for decryption. User A publishes their public key, while keeping the private key secret. Anyone can encrypt a message using A's public key, but only A can decrypt it with their private key.
The increased computation in asymmetric encryption cuts channel capacity, so it is mainly used in hybrid systems for authentication and digital signatures, or to exchange session keys.
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What are public keys and private keys used for?
In public-key cryptography, public and private keys serve distinct but complementary roles. Public keys are distributed widely and used to encrypt messages, ensuring that only the holder of the corresponding private key can decrypt and read them. Private keys, kept secret by the owner, are used to decrypt messages encrypted with the public key. This system enables secure communication without the need to exchange secret keys beforehand. Additionally, private keys can create digital signatures, verifying the authenticity and integrity of a message.
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How does public-key cryptography help keep online communication safe?
Public-key cryptography is an asymmetric form of cryptography that uses different keys for message transmission and reception, enhancing online communication security. Each user has a public key for encryption and a private key for decryption.
The sender encrypts the message using the recipient's public key, and only the recipient can decrypt it with their private key. This method ensures that even if the public key is intercepted, the message remains secure because the private key is required for decryption. This system promotes trust and security in online interactions.
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What are some common uses of public-key cryptography in everyday life?
Public-key cryptography, or asymmetric cryptography, is present in many facets of modern life. It uses two keys: a public key for encryption and a private key for decryption.
Examples of everyday applications:
* Securing online transactions Public-key cryptography is essential for secure e-commerce, protecting financial and personal data during online shopping and banking.
* Email encryption It ensures the confidentiality of email communications, preventing unauthorized access to sensitive information.
* VPNs Virtual private networks use public-key cryptography to create secure connections over the internet, protecting data from cyber threats.
* Digital signatures It provides a way to verify the authenticity and integrity of digital documents, ensuring that they haven't been tampered with.
* Smart cards Smart cards employ cryptographic protocols to secure transactions and verify user identity, enhancing security compared to traditional cards.3 Britannica SourcesCreated from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
public-key cryptography, asymmetric form of cryptography in which the transmitter of a message and its recipient use different keys (codes), thereby eliminating the need for the sender to transmit the code and risk its interception. It is commonly used in cryptocurrency transactions.
In 1976, in one of the most inspired insights in the history of cryptology, Sun Microsystems, Inc., computer engineer Whitfield Diffie and Stanford University electrical engineer Martin Hellman realized that the key distribution problem could be almost completely solved if a cryptosystem, T (and perhaps an inverse system, T′), could be devised that used two keys and satisfied the following conditions:
- It must be easy for the cryptographer to calculate a matched pair of keys, e (encryption) and d (decryption), for which TeT′d = I. Although not essential, it is desirable that T′dTe = I and that T = T′. Since most of the systems devised to meet points 1–4 satisfy these conditions as well, it will be assumed they hold hereafter—but that is not necessary.
- The encryption and decryption operation, T, should be (computationally) easy to carry out.
- At least one of the keys must be computationally infeasible for the cryptanalyst to recover even when he knows T, the other key, and arbitrarily many matching plaintext and ciphertext pairs.
- It should not be computationally feasible to recover x given y, where y = Tk(x) for almost all keys k and messages x.
Given such a system, Diffie and Hellman proposed that each user keep his decryption key secret and publish his encryption key in a public directory. Secrecy was not required, either in distributing or in storing this directory of “public” keys. Anyone wishing to communicate privately with a user whose key is in the directory only has to look up the recipient’s public key to encrypt a message that only the intended receiver can decrypt. The total number of keys involved is just twice the number of users, with each user having a key in the public directory and his own secret key, which he must protect in his own self-interest. Obviously the public directory must be authenticated, otherwise A could be tricked into communicating with C when he thinks he is communicating with B simply by substituting C’s key for B’s in A’s copy of the directory. Since they were focused on the key distribution problem, Diffie and Hellman called their discovery public-key cryptography. This was the first discussion of two-key cryptography in the open literature. However, Admiral Bobby Inman, while director of the U.S. National Security Agency (NSA) from 1977 to 1981, revealed that two-key cryptography had been known to the agency almost a decade earlier, having been discovered by James Ellis, Clifford Cocks, and Malcolm Williamson at the British Government Code Headquarters (GCHQ).
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In this system, ciphers created with a secret key can be decrypted by anyone using the corresponding public key—thereby providing a means to identify the originator at the expense of completely giving up secrecy. Ciphers generated using the public key can only be decrypted by users holding the secret key, not by others holding the public key—however, the secret-key holder receives no information concerning the sender. In other words, the system provides secrecy at the expense of completely giving up any capability of authentication. What Diffie and Hellman had done was to separate the secrecy channel from the authentication channel—a striking example of the sum of the parts being greater than the whole. Single-key cryptography is called symmetric for obvious reasons. A cryptosystem satisfying conditions 1–4 above is called asymmetric for equally obvious reasons. There are symmetric cryptosystems in which the encryption and decryption keys are not the same—for example, matrix transforms of the text in which one key is a nonsingular (invertible) matrix and the other its inverse. Even though this is a two-key cryptosystem, since it is easy to calculate the inverse to a non-singular matrix, it does not satisfy condition 3 and is not considered to be asymmetric.
Since in an asymmetric cryptosystem each user has a secrecy channel from every other user to him (using his public key) and an authentication channel from him to all other users (using his secret key), it is possible to achieve both secrecy and authentication using superencryption. Say A wishes to communicate a message in secret to B, but B wants to be sure the message was sent by A. A first encrypts the message with his secret key and then superencrypts the resulting cipher with B’s public key. The resulting outer cipher can only be decrypted by B, thus guaranteeing to A that only B can recover the inner cipher. When B opens the inner cipher using A’s public key he is certain the message came from someone knowing A’s key, presumably A. Simple as it is, this protocol is a paradigm for many contemporary applications.
Cryptographers have constructed several cryptographic schemes of this sort by starting with a “hard” mathematical problem—such as factoring a number that is the product of two very large primes—and attempting to make the cryptanalysis of the scheme be equivalent to solving the hard problem. If this can be done, the cryptosecurity of the scheme will be at least as good as the underlying mathematical problem is hard to solve. This has not been proven for any of the candidate schemes thus far, although it is believed to hold in each instance.
However, a simple and secure proof of identity is possible based on such computational asymmetry. A user first secretly selects two large primes and then openly publishes their product. Although it is easy to compute a modular square root (a number whose square leaves a designated remainder when divided by the product) if the prime factors are known, it is just as hard as factoring (in fact equivalent to factoring) the product if the primes are unknown. A user can therefore prove his identity, i.e., that he knows the original primes, by demonstrating that he can extract modular square roots. The user can be confident that no one can impersonate him since to do so they would have to be able to factor his product. There are some subtleties to the protocol that must be observed, but this illustrates how modern computational cryptography depends on hard problems.
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