# Public Key Encryption (PKE)

A Public Key Encryption (PKE) scheme is a primitive that allows someone to publish a public key that can be used to encrypt plaintext. Then, the generator of the key can use a secret key to decrypt the ciphertext. It is a critical notion for the modern public-key infrastructure and was famously made possible by [RSA78] and [Elg85].

## Formal Definition

### Syntax

A Public Key Encryption (PKE) scheme is a tuple of efficient functions , with respect to a keyspace , plaintext space , and ciphertext space , such that:

- , is a randomized function that takes a security parameter, and outputs a public key and secret key
- , is a randomized function that takes a public key and plaintext message , and outputs a ciphertext ,
- , is a deterministic function that takes a secret key and ciphertext , and outputs a plaintext message .

### Correctness

A PKE scheme is *correct* if for all , there exists a negligible function , such that

where .

### Chosen Plaintext Attack (CPA) Security

A PKE scheme is *CPA-secure* if for all efficient adversaries , there exists a negligible function , such that

where , is a uniformly random bit, and represents the state of the adversary after the first phase.

### Chosen Ciphertext Attack (CCA) Security

A SE scheme is *CCA-secure* if for all admissible efficient adversaries , there exists a negligible function , such that

where , is a uniformly random bit, and represents the state of the adversary after the first phase. Further, we say an adversary is admissible if it never queries on the ciphertext given to in the second phase of the game.

The difference in this definition is that we additionally give the adversary access to a decryption oracle. This is a strictly stronger definition than PKE#CPA-security.

## Relationship to other primitives

## Sufficient assumptions

See the sufficient assumptions for TDPs.

## Variations

## Other Notes

- Other security definitions, named CCA1 and CCA2, exist and have historically been confused with the CCA notion outlined above