Linear Algebra Problem Equivalencies

Reading:  First part of AT^2 handout for description bounded fan-in 
circuit model.

Theorem:  Let p be a prime number.  Then -1 is a quadratic residue mod p 
if and only if p = 2 or p = 1 mod 4.
Note: "-1 is a quadratic residue mod p" means p | x^2 + 1 for some 
integer x.

Problem A(n):  The input n bits represent the binary bits of a number p which
is prime and/or is even. 
 Here any prime less than 2^n may be represented as the leading bit(s) 
may be 0.  Return 1 if -1 is a quadratic residue mod p, else return 0.

Problem B(n):  The input n bits represent the binary bits of an n-bit 
prime p.  Here, for n > 1, the prime satisfies 2^{n-1} < p < 2^n as the leading bit 
is set to 1. Return 1 if -1 is a quadratic residue mod p, else return 0.

Note: algorithms to solve these problem are allowed to assume the n 
input bits denote a prime
and the algorithm is not responsible for what happens if that input 
condition is false.

Homework exercises:

1. In the bounded fan-in circuit model, we know that if a problem's 
result is not a trivial
function of n variables, then the parallel run time is at least log(n). 
Which of the above two problems is a trivial function of the input bits? 
 Give an O(1) time parallel algorithm for that
problem.

2. Some parallel models allow solutions to non-trivial problems in less 
than O(log(n)) time.
For instance consider a CRCW PRAM (syncronous processors have 
simultaneous access to a shared memory, in the case of concurrent writes 
to a single memory location in a given cycle, an arbitrary one of the 
writing processes succeeds in writing it's value there).  Show that the 
other problem (the one not a trivial function of the input bits) can be 
solved in O(1) time on a CRCW PRAM.


Advertisement:  For solving your exact linear algebra problems (eventually
in parallel) see the LinBox library at www.linalg.org! 

Questions? wan@cis.udel.edu