(2.1)
p. 106 #20.
a) Since
, we can apply Euler's Theorem, p. 81, to conclude that
. Hence
is an
with the property that
. Since
is, by definition, the smallest such value of
, it follows that
.
b)
=
=
=
= 1 mod n.
c) 1 =
=
=
=
.
d) Since by definition
is the smallest
positive
exponent of
to give 1, and
since
,
it follows from c)
that
is zero.
e) Taking
, and dividing by
, we see from part d) that the remainder
is zero. Hence

.
p. 106 #21.
isprime(601);
true
a) If
r  600 then the factorization of r contains only powers of the primes 2, 3, and
5. If r < 600, then at least one of the exponents of 2, 3, or 5 in the factorization of r
must be less than the corresponding exponent in the factorization of 600. The
maximal such elements are 300 where the exponent of 2 drops by 1; 200, where
the exponent of 3 drops by 1; and 120, where the exponent of 5 drops by 1 from
the exponents in the factorization in 600.
b) 601 is prime, hence
= 600. By #20,
 600. If
< 600,
then
must divide one of the maximal proper divisors, 300, 200, or 120.
c) By #20 b), if ord[601](7) divided 300, then
would be 1 mod 601, a
contradiction. Similarly for the other two cases.
d) Since by c)
does not divide any of the proper subdivisors of
,
but by #20 e) must divide
, then
.
e) If
for some
, so that 2 is not primitive,
and t is any
positive integer, then
=
=
=
, so
is also not primitive. Hence,
there is no point in testing for primitivity powers of integers where the original
integer is known not to be primitive.
We cannot exclude 6, even if 2 and 3 are known not to be primitive, since the
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(4.1)
exponents at which the 2 and the 3 become congruent to 1 may be different, so
that the product of the powers itself is different than 1 at each stage.
Once you know the order of 2 and the order of 3, however, you can immediately
discern the order of 6, since it is the least common multiple of the orders of 2 and
of 3. In fact, since the lcm(n,m) is often greater than both n and m, it is possible
that the composite 6 will have maximal order and 6 will be a primitive root of
unity although neither 2 or 3 is primitive.
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 Spring '08
 Staff
 Cryptography, Exponentiation, Power of two, ince

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