# 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, ${\displaystyle m}$, and a signature (or MAC), ${\displaystyle \sigma }$, that is valid for ${\displaystyle m}$, where ${\displaystyle m}$ 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 unforgeability, EUF)

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

#### Signature of a product of two messages

Take an algorithm, like RSA, with the multiplicative property:

${\displaystyle \sigma (m1)\cdot \sigma (m2)=\sigma (m1\cdot m2)}$.

This property can be exploited sending a message ${\displaystyle m'=m1\cdot m2}$ with a signature ${\displaystyle \sigma (m')=\sigma (m1\cdot m2)}$.[3]

A common defense to this attack is to hash the messages before signing them.[3]

### Selective forgery (selective unforgeability, SUF)

Selective forgery is the creation (by an adversary) of a message/signature pair ${\displaystyle (m,\sigma )}$ where ${\displaystyle m}$ has been chosen by the challenger prior to the attack.[4] ${\displaystyle m}$ may be chosen to have interesting mathematical properties with respect to the signature algorithm; however, in selective forgery, ${\displaystyle m}$ 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 unforgeability, UUF)

Universal forgery is the creation (by an adversary) of a valid signature, ${\displaystyle \sigma }$, for any given message, ${\displaystyle m}$. 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

1. ^ 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.
2. ^ Goldwasser, Shafi; Bellare, Mihir (2008). Lecture Notes on Cryptography. Summer course on cryptography. p. 170.
3. ^ a b Fabrizio d'Amore (April 2012). "Digital signatures - DSA" (PDF). La Sapienza University of Rome. pp. 8–9. Retrieved July 27, 2018.
4. ^ Smart, Nigel P. Cryptography Made Simple. Springer. p. 217. ISBN 978-3-319-21935-6.