FINDING N-TH ROOTS IN NILPOTENT GROUPS AND ...

FINDING N-TH ROOTS IN NILPOTENT GROUPS... 591
they both agree on a key. The security of this scheme lies on the difficulty of
finding square roots in 2-Engel groups. (see [7] for more information about this
part.)
One could extend this scheme to other n-Engel groups, since there are
similar relations in them as well (see [17]).
9.2. A Digital Signature Based on 4-Engel Groups
Consider a 4-Engel group, which is nilpotent and satisfies the following semigroup
law:
xy 2 xyx 2 y 2 x 2 yxy 2 xyx 2 yxy 2 x 2 y 2 xyx 2 y = yx 2 yxy 2 x 2 y 2 xyx 2 yxy 2 xyx 2 y 2 x 2 yxy 2 x
The idea to make a digital signature is as follows: Suppose x and y are
secret and x 2 , y 2 , xy 2 x and xy 2 x 2 y 2 x public information.
The public key is x 2 and the signature is the tuple of xy 2 x and xy 2 x 2 y 2 x.
The verifier knows y, so he only needs to verify both of the semigroup identity.
The security of this digital signature lies on the fact that the complexity of
finding square root in a 4-Engel group is experimentally proved to be exponential.
The Hirsch length of a 4−Engel group could even be 88 and nilpotent of
class 9 (see Werner Nickel’s webpage and [12]) and by the best known algorithm
by Nickel, we know that the complexity of finding matrix representations in a
nilpotent group of class 9 and Hirsch length 88 is exponential; which ultimately
gives us a solution to finding a square root in the underlying 4-engel group.
Acknowledgments
The authors are grateful to Professor Michael Anshel for helpful discussion
in the early stage of this project. The authors also thank Professor Gilbert
Baumslag for his useful comments. Delaram Kahrobaei, would like to thank
Professor Derek Holt for his helpful comments during her visit to Warwick University.
The research of Delaram Kahrobaei has been supported by a grant from
the City Tech Foundation and PSC CUNY Research Foundation of CUNY. The
research of Makenson and Renald has been supported by City Tech Foundation
and NSF-AMP.