Public-key cryptography: Difference between revisions

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[[File:Public key shared secret.svg|thumb|250px|right| In the [[Diffie–Hellman key exchange]] scheme, each party generates a public/private key pair and distributes the public key of the pair. After obtaining an authentic (n.b., this is critical) copy of each other's public keys, Alice and Bob can compute a shared secret offline. The shared secret can be used, for instance, as the key for a [[symmetric cipher]].]]

[[File:Public key encryption.svg|thumb|250px|right|In an asymmetric key encryption scheme, anyone can encrypt messages using a public key, but only the holder of the paired private key can decrypt such a message. The security of the system depends on the secrecy of the private key, which must not become known to any other.]]

'''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'''.{{Ref RFC|4949|notes=no}}<ref>{{Cite journal |last1=Bernstein |first1=Daniel J. |last2=Lange |first2=Tanja |date=2017-09-14 |title=Post-quantum cryptography |url=http://www.nature.com/articles/nature23461 |journal=Nature |language=en |volume=549 |issue=7671 |pages=188–194 |doi=10.1038/nature23461 |pmid=28905891 |bibcode=2017Natur.549..188B |s2ciымЗЩds2cid=4446249 |issn=0028-0836}}</ref> Key pairs are generated with [[cryptographic]] [[algorithms]] based on [[mathematical]] problems termed [[one-way function]]s. Security of public-key cryptography depends on keeping the private key secret; the public key can be openly distributed without compromising security.<ref>{{Cite book|url=https://books.google.com/books?id=Dam9zrViJjEC|title=Cryptography and Network Security: Principles and Practice|last=Stallings|first=William|date=3 May 1990|publisher=Prentice Hall|isbn=9780138690175|page=165|language=en}}</ref> There are many kinds of public-key cryptosystems, with different security goals, including [[digital signature]], [[Diffie–Hellman key exchange]], [[Key encapsulation mechanism|public-key key encapsulation]], and public-key encryption.

Public key algorithms are fundamental security primitives in modern [[cryptosystem]]s, including applications and protocols that offer assurance of the confidentiality and authenticity of electronic communications and data storage. They underpin numerous Internet standards, such as [[Transport Layer Security|Transport Layer Security (TLS)]], [[SSH]], [[S/MIME]], and [[Pretty Good Privacy|PGP]]. Compared to [[symmetric cryptography]], public-key cryptography can be too slow for many purposes,<ref>

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== Description ==

Before the mid-1970s, all cipher systems used [[symmetric key algorithm]]s, 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 cryptosystem, the public keys can be disseminated widely and openly, and only the corresponding private keys need be kept secret.

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== Applications ==

The most obvious application of a public key encryption system is for encrypting communication to provide [[confidentiality]] – a message that a sender encrypts using the recipient's public key, which can be decrypted only by the recipient's paired private key. Most digital services such as financial services, email, and messaging applications utilized daily are secured using public key encryption. <ref>{{Cite web |date=2025-06-05 |title=Post-Quantum Cryptography: A New Security Paradigm for the Post-Quantum Era |url=https://www.pentasecurity.com/blog/security-issue-post-quantum-cryptography/ |access-date=2025-07-10 |website=Penta Security Inc. |language=en-US}}</ref>

Another application in public key cryptography is the [[digital signature]]. Digital signature schemes can be used for sender [[authentication]].

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All public key schemes are in theory susceptible to a "[[brute-force attack|brute-force key search attack]]".<ref>{{cite book|last1=Paar|first1=Christof|first2=Jan|last2=Pelzl|first3=Bart|last3=Preneel|url=http://www.crypto-textbook.com|title=Understanding Cryptography: A Textbook for Students and Practitioners|publisher=Springer|year=2010|isbn=978-3-642-04100-6}}</ref> However, such an attack is impractical if the amount of computation needed to succeed – termed the "work factor" by [[Claude Shannon]] – 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 (algorithm)|RSA]] and [[ElGamal encryption]] have known attacks that are much faster than the brute-force approach.{{cn|date=June 2024}} None of these are sufficiently improved to be actually practical, however.

Major weaknesses have been found for several formerly promising asymmetric key algorithms. The [[Merkle–Hellman knapsack cryptosystem|"knapsack packing" algorithm]] was found to be insecure after the development of a new attack.<ref>{{Cite book|last1=Shamir|first1=Adi|title=23rd Annual Symposium on Foundations of Computer Science (SFCS 1982) |chapter=A polynomial time algorithm for breaking the basic Merkle-Hellman cryptosystem |date=November 1982 |chapter-url=https://ieeexplore.ieee.org/document/4568386 |pages=145–152|doi=10.1109/SFCS.1982.5}}</ref> As with all cryptographic functions, public-key implementations may be vulnerable to [[side-channel attack]]s 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 ===