Digital signature forgery

In a cryptographic digital signature or MAC system, digital signature forgery is the ability to create a pair consisting of a message, $m$ , and a signature (or MAC), $\sigma$ , that is valid for $m$ , where $m$ has not been signed in the past by the legitimate signer. There are three types of forgery: existential, selective, and universal.^[1]

Types

Besides the following attacks, there is also a total break: when adversary can compute the signer's private key and therefore forge any possible signature on any message.^[2]

Existential forgery

Existential forgery is the creation (by an adversary) of at least one message/signature pair, $(m, \sigma)$ , where $\sigma$ was not produced by the legitimate signer. The adversary need not have any control over $m$ ; $m$ need not have any particular meaning; the message content is irrelevant — as long as the pair, $(m, \sigma)$ , is valid, the adversary has succeeded in constructing an existential forgery.

Existential forgery is essentially the weakest adversarial goal, therefore the strongest schemes are those that are existentially unforgeable. Nevertheless, many state-of-art signature algorithms allow existential forgery. For example, an RSA forgery can be done as follows:

Let $e$ be the RSA public key.
Choose a random signature, $\sigma$ .
Send the message as: $\sigma ^{e}(\operatorname {mod} n)\parallel \sigma (\operatorname {mod} n)$ .
The recipient checks the signature: $\sigma ^{e}=\sigma ^{e}$ so the check will pass.

Note: The sender cannot control the message content so it will be a random message, that may help in some cases.

Multiplication forgery

This forgery can be used with two messages and their signatures as follows:

Let $\sigma _{1}=S_{k}(m_{1})$ be the RSA signature on the message, $m_{1}$ , under the key, $k$ .
Analogously, $\sigma _{2}=S_{k}(m_{2})$ .
In that case $\sigma _{1}\cdot \sigma _{2}{\pmod {n}}$ will be the valid RSA signature on the message, $m_{1}\cdot m_{2}{\pmod {n}}$ , under the key, $k$ .^[3]

Selective forgery

Selective forgery is the creation (by an adversary) of a message/signature pair $(m, \sigma)$ where $m$ has been chosen by the adversary prior to the attack. $m$ may be chosen to have interesting mathematical properties with respect to the signature algorithm; however, in selective forgery, $m$ must be fixed before the start of the attack.

The ability to successfully conduct a selective forgery attack implies the ability to successfully conduct an existential forgery attack.

Universal forgery

Universal forgery is the creation (by an adversary) of a valid signature, $\sigma$ , for any given message, $m$ . An adversary capable of universal forgery is able to sign messages he chose himself (as in selective forgery), messages chosen at random, or even specific messages provided by an opponent.

References

↑ Vaudenay, Serge (September 16, 2005). A Classical Introduction to Cryptography: Applications for Communications Security (1st ed.). Springer. p. 254. ISBN 978-0-387-25464-7.
↑ Goldwasser, Shafi; Bellare, Mihir (2008). Lecture Notes on Cryptography. Summer course on cryptography. p. 170.
↑ Kantarcioglu, Murat. "Digital Signatures" (PDF).

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