The ring End(Zp  Zp2 ). Characterization, extension and cryptographic applications

**Hora de inicio:** 12:00. **Hora de fin:** 13:00.

For a prime number p, Bergman (Israel Journal of Mathematics, 18: 257{277,

1974) established that End(Zp  Zp2 ) is a semilocal ring with p5 elements that

cannot be embedded in matrices over any commutative ring. We identify the

elements of End(ZpZp2 ) with elements in a new set, denoted by Ep, of matrices

of size 2  2, whose elements in the rst row belong to Zp and the elements

in the second row belong to Zp2 . By using polynomial functions over Ep we

introduce some key exchange protocols. These protocols are insecure because

all the elements of Ep are invertible. To avoid this weakness while maintaining

the good properties of Ep, we introduce a new noncommutative ring called E(m)

p ,

for some integer m  2, so that Ep is a subring of E(m)

p . This new ring has

the property that almost all the elements of E(m)

p are noninvertible for a good

choice of the parameters p and m.