Cryptography

LTO Network uses SHA256 to create cryptographic hashes. The BLAKE2b and SHA256 hashing algorithms are used for creating public/secret key pairs.
The ED25519 scheme is applied to create and verify signatures. X25519 with Salsa20 + Poly1305 is used for asymmetric encryption.
Base58 is used to create a string from an array of bytes.
If you want to create an application, you should find the implementation of these algorithms in your programming language.
Encoding
Base58
Arrays of bytes (like strings and hashes) in the project are encoded by Base58 algorithm with Bitcoin alphabet.
When encoding a hash, use the raw bytes and not the hexadecimal notation.
Note that encoding is not encryption. You can decode a base58 encoded message without the need of any type of key.
Examples
The string hello world is encoded as StV1DL6CwTryKyV.
SHA256 hash e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855 is encoded as
GKot5hBsd81kMupNCXHaqbhv3huEbxAFMLnpcX2hniwn 
Hashing
SHA256
In general Live Contracts uses the 256-bit version of SHA-2 for encoding. SHA-2 is one of the most common hashing algorithms which means it's available for almost all platforms and programming languages.
BLAKE2b + SHA256
When secure hashes are required, a combination of BLAKE2b and SHA256 are used.
BLAKE2b is supported by NaCl compatible libraries like libsodium as 'Generic hashing'. It's widely supported across platforms and languages.
Signing
ED25519
The default method for signing is using ED25519. This is an EdDSA algorithm on curve25519. It is designed to be faster than existing digital signature schemes without sacrificing security.
ED25519 is well supported on across many platforms and languages.
Functions for ED25519 are defined as sign in libsodium and nacl.
There are many valid (not unique!) signatures for one message when using ED25519. Also, you should not rely on any information before the hash and/or signature are checked.
secp256k1
Bitcoin, Ethereum, and many other blockchains use ECDSA with the secp256k1 curve for signing transactions. Outside of the realm of blockchain, this curve is barely used and not well supported.
secp256r1
The most commonly used and well-supported Elliptic Curve is NIST P-256. This is an ECDSA method using the secp256p1 curve.
Encryption
X25519
There are 3 algorithms involved for asymmetric encryption:
Key exchange: X25519
Encryption: XSalsa20 stream cipher
Authentication: Poly1305 MAC
This is a public key encryption schema, where the public key is used to encrypt data and the private key is used to decrypt data.
X25519 is sometimes referred to as Curve25519.
Functions that use the combination of these algorithms are defined as box in libsodium and nacl. 

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Encoding

Base58

Hashing

SHA256

BLAKE2b + SHA256

Signing

ED25519

secp256k1

secp256r1

Encryption

X25519