# Merkle Signature Scheme - Key Generation

Key Generation

The Merkle Signature Scheme can only be used to sign a limited number of messages with one public key . The number of possible messages must be a power of two, so that we denote the possible number of messages as .

The first step of generating the public key is to generate the public keys and private keys of one-time signatures. For each public key, with, a hash value is computed. With these hash values a hash tree is built.

We call a node of the tree, where denotes the level of the node. The level of a node is defined by the distance from the node to a leaf. Hence, a leaf of the tree has level and the root has level . We number all nodes of one level from the left to the right, so that is the leftmost node of level .

In the Merkle Tree the hash values are the leaves of a binary tree, so that . Each inner node of the tree is the hash value of the concatenation of its two children. So and .

In this way, a tree with leaves and nodes is built. The root of the tree is the public key of the Merkle Signature Scheme.

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Key Generation

Key generation is the process of generating keys for cryptography. A key is used to encrypt and decrypt whatever data is being encrypted/decrypted.

Modern cryptographic systems include symmetric-key algorithms (such as DES and AES) and public-key algorithms (such as RSA). Symmetric-key algorithms use a single shared key; keeping data secret requires keeping this key secret. Public-key algorithms use a public key and a private key. The public key is made available to anyone (often by means of a digital certificate). A sender encrypts data with the public key; only the holder of the private key can decrypt this data.

Since public-key algorithms tend to be much slower than symmetric-key algorithms, modern systems such as TLS and SSH use a combination of the two: one party receives the other's public key, and encrypts a small piece of data (either a symmetric key or some data used to generate it). The remainder of the conversation uses a (typically faster) symmetric-key algorithm for encryption.

Computer cryptography uses integers for keys. In some cases keys are randomly generated using a random number generator (RNG) or pseudorandom number generator (PRNG). A PRNG is a computer algorithm that produces data that appears random under analysis. PRNGs that use system entropy to seed data generally produce better results, since this makes the initial conditions of the PRNG much more difficult for an attacker to guess. In other situations, the key is created using a passphrase and a key generation algorithm, usually involving a cryptographic hash function such as SHA-1.

The simplest method to read encrypted data is a brute force attack—simply attempting every number, up to the maximum length of the key. Therefore, it is important to use a sufficiently long key length; longer keys take exponentially longer to attack, rendering a brute force attack impractical. Currently, key lengths of 128 bits (for symmetric key algorithms) and 1024 bits (for public-key algorithms) are common.

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