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The Merkle signature scheme is a digital signature scheme based on hash trees (also called Merkle trees) and one-time signatures such as the Lamport signature scheme. It was developed by Ralph Merkle in the late 70s and is an alternative to traditional digital signatures such as the Digital Signature Algorithm or RSA. The advantage of the Merkle Signature Scheme is, that it is believed to be resistant against quantum computer algorithms. The traditional public key algorithms, such as RSA and ELGamal would become insecure in case an effective quantum computer can be built (Shor's algorithm). The Merkle Signature Scheme however only depends on the existence of secure hash functions. This makes the Merkle Signature Scheme very adjustable and resistant against quantum computing.

## Key generationEdit

The Merkle Signature Scheme can only be used to sign a limited number of messages with one public key $pub$. The number of possible messages must be a power of two, so that we denote the possible number of messages as $N=2^n$.

The first step of generating the public key $pub$ is to generate the public keys $X_i$ and private keys $Y_i$ of $2^n$ one-time signatures. For each private key $Y_i$, with $1 \leq i \leq 2^n$, a hash value $h_i=H(Y_i)$ is computed. With these hash values $h_i$ a hash tree is built.

We call a node of the tree $a_{i,j}$, where $i$ 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 $i=0$ and the root has level $i=n$. We number all nodes of one level from the left to the right, so that $a_{i,0}$ is the leftmost node of level $i$.

In the Merkle Tree the hash values $h_i$ are the leafs of a binary tree, so that $h_i=a_{0,i}$. Each inner node of the tree is the hash value of the concatenation of its two children. So $a_{1,0}=H(a_{0,0}||a_{0,1})$ and $a_{2,0}=H(a_{1,0}||a_{1,1})$. An example of a merkle tree is illustrated in figure \ref{fig:gra1}.

In this way, a tree with $2^n$ leafs and $2^{n+1}-1$ nodes is built. The root of the tree $a_{n,0}$ is the public key $pub$ of the Merkle Signature Scheme.

## Signature generationEdit

To sign a message $M$ with the Merkle Signature Scheme, the message $M$ is signed with a one-time signature scheme, resulting in a signature $sig'$, first. This is done, by using one of the public and private key pairs $(X_i, Y_i,)$.

The corresponding leaf of the hash tree to a one-time public key $X_i$ is $a_{0,i}=H(Y_i)$. We call the path in the hash tree from $a_{0,i}$ to the root $A$. The path $A$ consists of $n+1$ nodes, $A_0, ... A_n$, with $A_0=a_{0,i}$ being the leaf and $A_n=a_{n,0}=pub$ being the root of the tree. To compute this path $A$, we need every child of the nodes $A_{1}, ..., A_{n}$. We know that $A_i$ is a child of $A_{i+1}$. To calculate the next node $A_{i+1}$ of the path $A$, we need to know both children of $A_{i+1}$. So we need the brother node of $A_i$. We call this node $auth_i$, so that $A_{i+1}=H(A_i||auth_i)$. Hence, $n$ nodes $auth_0,...,auth_{n-1}$ are needed, to compute every node of the path $A$. We now calculate and save these nodes $auth_{0}, ... , auth_{n-1}$.

These nodes, plus the one-time signature $sig'$ of $M$ is the signature $sig=(sig'||auth_2||auth_3||...||auth_{n-1})$ of the Merkle Signature Scheme. An example of an authentication path is illustrated in figure \ref{fig:gra2}.

## Signature verificationEdit

The receiver knows the public key $pub$, the message $M$, and the signature $sig=(sig'||auth_0||auth_1||...||auth_{n-1})$. At first, the receiver verifies the one-time signature $sig'$ of the message $M$. If $sig'$ is a valid signature of $M$, the receiver computes $A_0=H(Y_i)$ by hashing the public key of the one-time signature. For $j=1,..,n-1$, the nodes of $A_j$ of the path $A$ are computed with $A_j=H(a_{j-1}||b_{j-1})$. If $A_n$ equals the public key $pub$ of the merkle signature scheme, the signature is valid.

## References Edit

• G. Becker. "Merkle Signature Schemes, Merkle Trees and Their Cryptanalysis", seminar 'Post Quantum Cryptology' at the Ruhr-University Bochum, Germany.
• E. Dahmen, M. Dring, E. Klintsevich, J. Buchmann, L.C. Coronado Garca. "CMSS - an improved merkle signature scheme". Progress in Cryptology - Indocrypt 2006, 2006.
• E. Klintsevich, K. Okeya, C.Vuillaume, J. Buchmann, E.Dahmen. "Merkle signatures with virtually unlimited signature capacity". 5th International Conference on Applied Cryptography and Network Security - ACNS07, 2007.
• Ralph Merkle. "Secrecy, authentication and public key systems / A certified digital signature". Ph.D. dissertation, Dept. of Electrical Engineering, Stanford University, 1979. [1]
• S. Micali, M. Jakobsson, T. Leighton, M. Szydlo. "Fractal merkle tree representation and traversal". RSA-CT 03, 2003