Exercise 1.28: One variant of the Fermat test
that cannot be fooled is called the 
Miller-Rabin test (Miller 1976;
Rabin 1980).  This starts from an alternate form of Fermat’s Little Theorem,
which states that if 
  n
 is a prime number and 
  a
 is any positive integer
less than 
  n
, then 
  a
 raised to the 
  
    (
    n
    −
    1
    )
  
-st power is congruent to 1
modulo 
  n
.  To test the primality of a number 
  n
 by the Miller-Rabin
test, we pick a random number 
  
    a
    <
    n
  
 and raise 
  a
 to the 
  
    (
    n
    −
    1
    )
  
-st
power modulo 
  n
 using the expmod procedure.  However, whenever we
perform the squaring step in expmod, we check to see if we have
discovered a “nontrivial square root of 1 modulo 
  n
,” that is, a number
not equal to 1 or 
  
    n
    −
    1
  
 whose square is equal to 1 modulo 
  n
.  It is
possible to prove that if such a nontrivial square root of 1 exists, then 
  n

is not prime.  It is also possible to prove that if 
  n
 is an odd number that
is not prime, then, for at least half the numbers 
  
    a
    <
    n
  
, computing

  
    a
    
      n
      −
      1
    
  
 in this way will reveal a nontrivial square root of 1 modulo

  n
.  (This is why the Miller-Rabin test cannot be fooled.)  Modify the
expmod procedure to signal if it discovers a nontrivial square root of
1, and use this to implement the Miller-Rabin test with a procedure analogous
to fermat-test.  Check your procedure by testing various known primes
and non-primes.  Hint: One convenient way to make expmod signal is to
have it return 0.