Asymmetric cryptography, also known as public key cryptography, uses public and private keys to encrypt and decrypt data. The keys are simply large numbers that have been paired together but are not identical (asymmetric). One key in the pair can be shared with everyone; it is called the public key. The other key in the pair is kept secret; it is called the private key. Either of the keys can be used to encrypt a message; the opposite key from the one used to encrypt the message is used for decryption.
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Many protocols like SSH, OpenPGP, S/MIME, and SSL/TLS rely on asymmetric cryptography for encryption and digital signature functions. It is also used in software programs, such as browsers, which need to establish a secure connection over an insecure network like the internet or need to validate a digital signature. Encryption strength is directly tied to key size and doubling key length delivers an exponential increase in strength, although it does impair performance. As computing power increases and more efficient factoring algorithms are discovered, the ability to factor larger and larger numbers also increases.
For asymmetric encryption to deliver confidentiality, integrity, authenticity and non-repudiability, users and systems need to be certain that a public key is authentic, that it belongs to the person or entity claimed and that it has not been tampered with or replaced by a malicious third party. There is no perfect solution to this public key authentication problem. A public key infrastructure (PKI), where trusted certificate authorities certify ownership of key pairs and certificates, is the most common approach, but encryption products based on the Pretty Good Privacy (PGP) model (including OpenPGP), rely on a decentralized authentication model called a web of trust, which relies on individual endorsements of the link between user and public key.
Whitfield Diffie and Martin Hellman, researchers at Stanford University, first publicly proposed asymmetric encryption in their 1977 paper, "New Directions in Cryptography." The concept had been independently and covertly proposed by James Ellis several years before, while working for the Government Communications Headquarters (GCHQ), the British intelligence and security organization. The asymmetric algorithm as outlined in the Diffie-Hellman paper uses numbers raised to specific powers to produce decryption keys.
RSA (Rivest-Shamir-Adleman), the most widely used asymmetric algorithm, is embedded in the SSL/TLS protocol which is used to provide communications security over a computer network. RSA derives its security from the computational difficulty of factoring large integers that are the product of two large prime numbers. Multiplying two large primes is easy, but the difficulty of determining the original numbers from the total -- factoring -- forms the basis of public key cryptography security. The time it takes to factor the product of two sufficiently large primes is considered to be beyond the capabilities of most attackers, excluding nation state actors who may have access to sufficient computing power. RSA keys are typically 1024- or 2048-bits long, but experts believe that 1024-bit keys could be broken in the near future, which is why government and industry are moving to a minimum key length of 2048-bits.
Elliptic Curve Cryptography (ECC) is gaining favor with many security experts as an alternative to RSA for implementing public-key cryptography. ECC is a public key encryption technique based on elliptic curve theory that can create faster, smaller, and more efficient cryptographic keys. ECC generates keys through the properties of the elliptic curve equation. To break ECC, one must compute an elliptic curve discrete logarithm, and it turns out that this is a significantly more difficult problem than factoring. As a result, ECC key sizes can be significantly smaller than those required by RSA yet deliver equivalent security with lower computing power and battery resource usage making it more suitable for mobile applications than RSA.
Digital signatures and asymmetric cryptography
Digital signatures are based on asymmetric cryptography and can provide assurances of evidence to origin, identity and status of an electronic document, transaction or message, as well as acknowledging informed consent by the signer. To create a digital signature, signing software (such as an email program) creates a one-way hash of the electronic data to be signed. The user's private key is then used to encrypt the hash, returning a value that is unique to the hashed data. The encrypted hash, along with other information such as the hashing algorithm, forms the digital signature. Any change in the data, even to a single bit, results in a different hash value. This attribute enables others to validate the integrity of the data by using the signer's public key to decrypt the hash. If the decrypted hash matches a second computed hash of the same data, it proves that the data hasn't changed since it was signed. If the two hashes don't match, the data has either been tampered with in some way (indicating a failure of integrity) or the signature was created with a private key that doesn't correspond to the public key presented by the signer (indicating a failure of authentication).
A digital signature also makes it difficult for the signing party to deny having signed something (the property of non-repudiation). If a signing party denies a valid digital signature, their private key has either been compromised, or they are being untruthful. In many countries, including the United States, digital signatures have the same legal weight as more traditional forms of signatures.