CMSC 414 Homework 2 Grading Key for Signature Schemes and Euler's Totient Function - Prof., Assignments of Computer Science

The grading key for homework 2 of cmsc 414, focusing on signature schemes and euler's totient function. It covers the calculation of signatures for messages and multiplication of signatures, as well as properties of euler's totient function.

Typology: Assignments

Pre 2010

Uploaded on 02/13/2009

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Fall 2006 CMSC 414: HW 2 Grading Key
Total 7 points
__________________________________________________________
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
1โˆ’1
) = s
1โˆ’1
mod-n , assuming m
1โˆ’1
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].
___________________________________________________________
7. [3 points]
1- writing something
2- a) ฯ†(p
a
) = (p-1)โ‹…p
aโˆ’1
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
โˆ’1)โ‹…p
1a1โˆ’1
โ‹… (p
2
โˆ’1)โ‹…p
2a2โˆ’1
โ‹…โ‹…โ‹… (p
k
โˆ’1)โ‹…p
kakโˆ’1
___________________________________________________________

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Fall 2006 CMSC 414: HW 2 Grading Key

Total 7 points


  1. [4 points] 1- writing something 2,3,4- a) Let s 1 be the signature of m 1 , i.e., s 1 = m 1 d^ mod-n. Let s 2 be the signature of m 2 , i.e., s 2 = m 2 d^ mod-n. Signature(m 1 j) = s 1 j^ mod-n Signature(m 1 โˆ’^1 ) = s 1 โˆ’^1 mod-n , assuming m 1 โˆ’^1 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 1 d)โ‹…(m 2 d) mod-n]. c) Signature(m 1 jโ‹…m 2 k) = s 1 jโ‹…s 2 k^ mod-n [from above].

  1. [3 points] 1- writing something 2- a) ฯ†(pa) = (p-1)โ‹…paโˆ’^1 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‹…, pn, q are distinct primes, then (p 1 a1โ‹…p 2 a2^ โ‹…โ‹…โ‹…โ‹… pnan) and qb^ are relatively prime. c) ฯ†( p 1 a1^ โ‹… p 2 a2^ โ‹… โ‹… โ‹… pkak^ ) = (p 1 โˆ’1)โ‹…p 1 a1โˆ’^1 โ‹… (p 2 โˆ’1)โ‹…p 2 a2โˆ’^1 โ‹…โ‹…โ‹… (pkโˆ’1)โ‹…pkakโˆ’^1