Merkle signature scheme

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 generation

Merkle Tree with 8 leaves

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 = 2ⁿ.

The first step of generating the public key pub is to generate the public keys X_i and private keys Y_i of 2ⁿ 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 leaves 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 2ⁿ leaves 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 generation

Merkle tree with path A and authentication path for i=2

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, A₀,...A_n, with A₀ = 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 A₁,...,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 auth₀,...,auth_{n − 1} are needed, to compute every node of the path A. We now calculate and save these nodes auth₀,...,auth_{n − 1}.

These nodes, plus the one-time signature sig' of M is the signature sig = (sig' | | auth₂ | | auth₃ | | ... | | auth_{n − 1}) of the Merkle Signature Scheme. An example of an authentication path is illustrated in the figure on the right.

Signature verification

The receiver knows the public key pub, the message M, and the signature sig = (sig' | | auth₀ | | auth₁ | | ... | | 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 A₀ = 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} | | auth_{j − 1}). If A_n equals the public key pub of the merkle signature scheme, the signature is valid.

References

• 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

External links

• Efficient Use of Merkle Trees - RSA labs explanation of the original purpose of Merkle trees + Lamport signatures, as an efficient one-time signature scheme.