Cryptography Reference
In-Depth Information
2
e.
x
12 (mod 69)
2
f.
8 (mod 217)
Check the solution(s) you obtain.
x
3.
Solve the following quadratic congruences:
a.
2
+ 2
3
x
x
0 (mod 7)
2
+ 3
b.
2
x
x
+ 9
5 (mod 7)
2
+ 10
c.
18 (mod 23)
Check the solution(s) you obtain.
5
x
x
+ 13
4.
Solve the following quadratic congruences:
a.
2
+ 2
3
x
x
12 (mod 77)
2
+ 3x + 9
b.
2
x
106 (mod 133)
2
+ 10
c.
101 (mod 209)
Check the solution(s) you obtain.
5
x
x
+ 13
5.
Prove proposition 30.
6.
Solve the following quadratic congruences:
a.
2
+ 2
4
x
x
+ 100
58 (mod 231)
2
+ 3
b.
0 (mod 1463)
Check the solution(s) you obtain.
2
x
x
+ 182
7.
The solveQuadratic() method can be written in a much “cleaner” way. First, write a
method to solve quadratic congruences of the form
x
2
a
(mod
p
)
where
is a prime congruent to 3 modulo 4. Use this method in conjunction with the
solveCRT() method, and use the Chinese Remainder Theorem to produce the solutions.
p
2
+
8.
Suppose the quadratic congruence
ax
bx
+
c
0 (mod
n
) has solutions, and that
n
=
p
i
is unique, and each congruent to 3 modulo 4.
Explain how you would find the solutions.
p
1
p
2
...
p
m
, where each prime factor
9.
Revise the solveQuadratic() method to compute and return solutions of quadratic con-
gruences as described in the previous exercise.
Search WWH ::
Custom Search