# Can a fast signature scheme without secret key be secure

# Un Schema de Signature Courte et Rapide N'Utilisant pas de CLE Secrete Peut-IL Etre Fiable?

## Abstract

Another title could have been "A probabilistic factorization algorithm in GL(2,p)". However, the problem is to calculate a fast and short signature associated with a plaintext inscribed on an erasable support. The signature should be written down in a book accompanying the record in order that it could be check ed anytime that the latter has not been changed. J. BOSSET [1] suggest such a scheme together with an algorithm for computing a signature. The 64 characters needed for the plaintext are identified with a subset of GL(2,p), p=997. The signature is the product of the matrices corresponding to the plaintext characters taken in the order where they appear. Such a scheme could be broken if it is possible to factorize an element of GL(2,p) into t=16 r factors, each one in a subset **U**_{i} of GL(2,p) of size 64 , i=1,...,t. We here assume one hypothesis only on uniform probability distributions of random variables defined on product sets **V**_{j}=**U**_{jr+1}×...×**U**_{(j+1)r}, j=0,...,15. In consideration on which, a probabilistic factorization algorithm in GL(2,p) is introduced.

It is shown that for p=10,007, drawing according to a uniform probability distribution a sequence of 11,952 elements in each **V**_{j} provides the whole needed material to factorizing with a probability of success of at least 97%. The most expensive operation in the algorithm is sorting each of the sequences.

## Keywords

Geometric Series Stirling Number Short Signature Bernoulli Trial Uniform Probability Distribution## Preview

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## References

- [1]J. BOSSET: "Contre les risques d'altération, un système de certification des informations", 01 Informatique n
^{o}107, Février 1977.Google Scholar - [2]W. FELLER: "An introduction to probability theory and its applications", Wiley, 1968.Google Scholar
- [3]L. COMTET: "Advanced combinatoris", D. Reidel, 1974.Google Scholar