When we work in finite algebra and take into account the 
computational complexity of homomorphisms between objects (groups, 
rings, etc.) -- as it is required in cryptography and computational 
algebra -- we find ourselves in a strange world where, most likely, 
almost nothing is invertible. However, this non-invertibility, in its 
more important manifestations (which includes famous problems such as 
the Discrete Logarithm Problem over finite fields) had so far never been 
proven.

But I will try to demonstrate that a sincere acceptance of this 
(thermodynamic in its nature) non-invertibility allows us to develop 
surprisingly efficient algorithms which solve previously inaccessible 
problems in computational group theory.

This is a joint work with Sukru Yalcinkaya; I will describe and 
informally explain  a few of our very recent results.