The RSA (Rivest–Shamir–Adleman) cryptosystem is a family of public-key cryptosystems, one of the oldest widely used for secure data transmission. The initialism "RSA" comes from the surnames of Ron Rivest, Adi Shamir and Leonard Adleman, who publicly described the algorithm in 1977. An equivalent system was developed secretly in 1973 at Government Communications Headquarters (GCHQ), the British signals intelligence agency, by the English mathematician Clifford Cocks. That system was declassified in 1997.

RSA is used in digital signature such as RSASSA-PSS or RSA-FDH, public-key encryption of very short messages (almost always a single-use symmetric key in a hybrid cryptosystem) such as RSAES-OAEP, and public-key key encapsulation.

In RSA-based cryptography, a user's private key—which can be used to sign messages, or decrypt messages sent to that user—is a pair of large prime numbers chosen at random and kept secret. A user's public key—which can be used to verify messages from the user, or encrypt messages so that only that user can decrypt them—is the product of the prime numbers.

The security of RSA is related to the difficulty of factoring the product of two large prime numbers, the "factoring problem". Breaking RSA encryption is known as the RSA problem. Whether it is as difficult as the factoring problem is an open question. There are no published methods to defeat the system if a large enough key is used.

The idea of an asymmetric public-private key cryptosystem is attributed to Whitfield Diffie and Martin Hellman, who published this concept in 1976. They also introduced digital signatures and attempted to apply number theory. Their formulation used a shared-secret-key created from exponentiation of some number, modulo a prime number. However, they left open the problem of realizing a one-way function, possibly because the difficulty of factoring was not well-studied at the time. Moreover, like Diffie-Hellman, RSA is based on modular exponentiation.

Ron Rivest, Adi Shamir, and Leonard Adleman at the Massachusetts Institute of Technology made several attempts over the course of a year to create a function that was hard to invert. Rivest and Shamir, as computer scientists, proposed many potential functions, while Adleman, as a mathematician, was responsible for finding their weaknesses. They tried many approaches, including "knapsack-based" and "permutation polynomials". For a time, they thought what they wanted to achieve was impossible due to contradictory requirements. In April 1977, they spent Passover at the house of a student and drank a good deal of wine before returning to their homes at around midnight. Rivest, unable to sleep, lay on the couch with a math textbook and started thinking about their one-way function. He spent the rest of the night formalizing his idea, and he had much of the paper ready by daybreak. The algorithm is now known as RSA – the initials of their surnames in same order as their paper.

Clifford Cocks, an English mathematician working for the British intelligence agency Government Communications Headquarters (GCHQ), described a similar system in an internal document in 1973. However, given the relatively expensive computers needed to implement it at the time, it was considered to be mostly a curiosity and, as far as is publicly known, was never deployed. His ideas and concepts were not revealed until 1997 due to its top-secret classification.

Kid-RSA (KRSA) is a simplified, insecure public-key cipher published in 1997, designed for educational purposes. Kid-RSA gives insight into RSA and other public-key ciphers, analogous to simplified DES.