This preview shows pages 1–2. Sign up to view the full content.
The University of Sydney
MATH2068
Number Theory and Cryptography
(http://www.maths.usyd.edu.au/u/UG/IM/MATH2068/)
Semester 2, 2008
Lecturer: R. Howlett
Tutorial 12
1.
You are given that 941 is prime.
(
i
)
Working with residue arithmetic mod 941, how many multiplications are
needed to compute 6
235
? (For example, three multiplications are needed
to compute 6
8
.)
(
ii
) Using a calculator, check (at least part of) the following table of powers of
6 (reduced mod 941).
i
2
4
8 16 32 64 128 192 224 232 234 235
6
i
36 355 872 56 313 105 674 195 811 501 157 1
(
iii
) Solve
x
2
≡
6 (mod 941). [Hint: ﬁnd an even
n
with 6
n
≡
6.]
(For example, to ﬁnd the residue of 1296 mod 941 using your calculator, divide 1296
by 941, subtract oﬀ the integer part, and multiply back by 941.)
Solution.
(
i
)
Twelve multiplications are needed: one for each entry in the table above.
One multiplication gives you 6
2
, you square that to get 6
4
, square that to get 6
8
,
and so on up to 6
128
, by which time you have done seven multiplications. Since
you have computed 6
64
and 6
128
you can multiply them to get 6
192
, multiply
that by 6
32
(already calculated) to get 6
224
, then multiply that by 6
8
, then by
6
2
, then by 6.
No solution is provided, or (I hope) needed for Part (
ii
). However, you are
hereby warned that such calculator calculations may be required in the exam.
(
iii
)
From the table we see that 6
470
= (6
235
)
2
= 1; so 6 is a square mod 941.
So a solution deﬁnitely exists. But – better still – the table shows us that 6 raised
to an odd power (namely, 235) equals 1. So 6
236
= 6, and the square roots of 6
must be
±
6
118
. From the table we see that 6
118
= 105
×
313
×
56
×
355
×
36,
and on calculating this we ﬁnd that it is 299. So the square roots of 6 are 299
and

299 = 642.
2.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 One '08
 Howlett
 Number Theory, Cryptography

Click to edit the document details