p. 242, #11.
p. 252, # 1, 2, 3, 4, 5, 6
p. 255, # 1, 2, 3.
p. 242 #11.
We assume that
and
are 512 bit numbers, satisfying h(IV,
) = h(IV,
).
A 512 bit numbers is always bounded above by
, so
and
(a) By the CRT, we know that there is a unique solution
to the simultaneous linear
congruences
and
.
Note that
<
<
.
(b)
The value of
is almost constant over the range from
to
, and
is
never much bigger than
. (The same applies to
.)
We see that if
is at least 1, then
<
. Moreover,
<
<
<
.
Now if
, then
<
=
. So for all k's in the range from 0 to
,
is
at most
of
, so
lies in the (proportionally) short interval between
1 and 1.25 times
, so the logarithm is essentially constant.
But since
<
<
<
, then
is at most
=
=
, which is about
in the logarithmic sense.