Public-Key Cryptography
Cryptography with paired public and private keys, enabling encryption and digital signatures without shared secrets.
Public-key (asymmetric) cryptography uses pairs of mathematically linked keys: a public key, which can be shared freely, and a private key, which must be kept secret. What one key encrypts, only the other decrypts. This solves the key-distribution problem of symmetric cryptography, where both parties must somehow share the same secret before communicating.
The field began in 1976 with the Diffie–Hellman key exchange, which showed how two parties can agree on a shared secret over an insecure channel, and in 1977 Rivest, Shamir, and Adleman published RSA, the first complete public-key cryptosystem. RSA's security rests on the difficulty of factoring large integers: the public key includes the product n = p·q of two large primes, and recovering the private key requires factoring n, which is believed infeasible for sufficiently large n — today 2048 to 4096 bits. The fundamental theorem of arithmetic guarantees that the factorization exists and is unique; finding it is the hard part.
Elliptic-curve cryptography (ECC) achieves comparable security with much shorter keys — a 256-bit ECC key is commonly compared with a 3072-bit RSA key — and is now the default for key exchange (X25519) and signatures (Ed25519). Digital signatures, the other core use of public-key systems, provide authenticity and integrity: a signature can be produced only with the private key but verified by anyone holding the public key.
Public-key cryptography secures HTTPS: TLS handshakes use it to authenticate servers and exchange symmetric session keys, after which the bulk of traffic is encrypted symmetrically for speed. Its practical vulnerabilities are rarely the mathematics — they are key management: leaked private keys, compromised certificate authorities, and weak randomness. Shor's algorithm would break RSA and ECC on a sufficiently powerful quantum computer, which is why post-quantum cryptography is an active standardization effort.
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cryptography encryption rsa security
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