First we assume that we have access to the stream of unassociated primes
*p*_{1}, *p*_{2}, *p*_{3},..., such that
(*p*_{1}) < (*p*_{1}*p*_{2}) < (*p*_{1}*p*_{2}*p*_{3}) < ^{ ... }.
Indeed the recovery of an element *a* in *E* from
*a* mod*m*=*p*_{1}^{ ... }*p*_{n}
requires sufficiently large *m*.

Secondly, we assume the avialability of a mapping
*scs*
from
*E*×*E* {0} to *E*, called a *symmetric canonical simplifier*,
such that we have the following properties.

**Simplification.**- Any element
*a**E*must satisfy*a**scs*(*a*,*m*) for any*m**E*{0}. More formally:( *a**E*)(*m**E*{0})*a**scs*(*a*,*m*).(116)

**Canonicity.**- For any
*m**E*{0}, any two elements*a*,*b**E*equivalent modulo*m*must satisfy*scs*(*a*,*m*) =*scs*(*b*,*m*). More formally:( *a*,*b**E*)(*m**E*{0}) (*a**b*) (*scs*(*a*,*m*) =*scs*(*b*,*m*)).(117)

**Recoverage = symmetry.**- All elements of a bounded degree are recovered
by the simplifier if the modulus is sufficiently large.
( *B*> 0)(*M*)( (*a*,*m*)*E*×*E*{0})*scs*(*a*,*m*) =*a*.(118)

See [KM99] for more details.

2004-04-27