Fall 2006 CMSC 414: HW 2 Grading Key
Total 7 points
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4. [4 points]
1- writing something
2,3,4- a) Let s
1
be the signature of m
1
, i.e., s
1
= m
1d
mod-n.
Let s
2
be the signature of m
2
, i.e., s
2
= m
2d
mod-n.
Signature(m
1j
) = s
1j
mod-n
Signature(m
1โ1
) = s
1โ1
mod-n , assuming m
1โ1
exists.
b) Signature(m
1
โ
m
2
) = s
1
โ
s
2
mod-n
[because (m
1
โ
m
2
)
d
mod-n = (m
1d
)โ
(m
2d
) mod-n].
c) Signature(m
1j
โ
m
2k
) = s
1j
โ
s
2k
mod-n [from above].
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7. [3 points]
1- writing something
2- a) ฯ(p
a
) = (p-1)โ
p
aโ1
for p prime and a > 0
ฯ(pโ
q) = ฯ(p)โ
ฯ(q) for p and q relatively prime
b) If p
1
, p
2
, โ
โ
โ
, p
n
, q are distinct primes,
then (p
1a1
โ
p
2a2
โ
โ
โ
โ
p
nan
) and q
b
are relatively prime.
c) ฯ( p
1a1
โ
p
2a2
โ
โ
โ
p
kak
) = (p
1
โ1)โ
p
1a1โ1
โ
(p
2
โ1)โ
p
2a2โ1
โ
โ
โ
(p
k
โ1)โ
p
kakโ1
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