# Public-Key Cryptography

Learn asymmetric cryptography in detail in this lesson.

## Asymmetric (public-key) cryptography

Public-key cryptography works quite differently than symmetric cryptography since there is not just one key that Bob and Alice share, but rather a pair of complementary keys that are mathematically related: a private key $k_{\text {priv }}$ that’s only known to its owner (Alice) and a public key $k_{\text {pub }}$, which she makes publicly available to everyone in order to serve as identification of her account. Thus, any person can encrypt and send a message against Alice’s public key, whereas only Alice is able to decrypt the encrypted ciphertext because she is the only one who is in possession of the matching private key $k_{\text {priv }}$.

However, all actual public-key cryptography systems depend heavily on cryptographic algorithms where no efficient solution is known. As we’ll see, this kind of algorithm relies on so-called one-way functions. This means that the security of a cryptosystem is based on the belief that specific number-theoretic functions are very hard to invert, whereas, we should notice that there’s no mathematical proof that these problems are

**Integer factorization schemes,**which base themselves on the hardness of factorization of large integers, such as the RSA public-key encryption and signature scheme ( ).Ronald L. Rivest et al. (1978) Ronald L. Rivest, Adi Shamir, and Leonard Adleman. A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM, 21(2): 120-126, February 1978. **Discrete logarithm schemes,**which rely on the difficulty of the discrete logarithm problem (DLP), such as the ElGamal public-key encryption and signature scheme or the Digital Signature Algorithm (DSA), which is essentially an extension of the ElGamal scheme.**Elliptic curve schemes,**which rely on the hardness of the elliptic curve discrete logarithm problem (ECDLP), such as the Elliptic Curve Digital Signature Algorithm (ECDSA) scheme.

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