Exercise 1.5 Primes, Powers And Square Roots Worksheet - Heinemann Maths Zone Page 7
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11Questions
1 Which two prime numbers were multiplied together to give each of these
encryption keys?
(a) 77
(b) 38
(c) 65
(d) 202
(e) 143
(f) 2183
2 Multiply the following prime numbers together to get encryption keys.
(a) 3 and 17
(b) 7 and 31
(c) 47 and 73
(d) 101 and 103
(e) 131 and 727
(f) 313 and 93 139
3 (a) Write out the first five prime numbers.
(b) Write out the six smallest keys that you can create by multiplying two primes.
(c) How many factors does each key have?
(d) Do you think every key created by multiplying two primes together will have this
number of factors? Why?
4 Here’s a simplified version of how decryption using prime numbers works. Suppose the
encrypted message is W L R G Y S B E M R K C R E B J C V and you know the key is 713.
You have to find the two prime numbers that multiply together to give 713. This is
reasonably easy for such a small number. The answer is 23 and 31.
These digits tell you how much the letters in the original message have been shifted
backwards along the alphabet to give the coded message.You match these digits to the
code as follows:
W
L
R
G
Y
S
B
E
M
R
K
C
R
E
B
J
C
V
2
3
3
1
2
3
3
1
2
3
3
1
2
3
3
1
2
3
The way decrypting this code works is that the number tells you how many letters to
shift each letter forwards along the alphabet. The 2 under the W means shift two
letters forward through the alphabet from W, so the uncoded message, usually called
the plaintext, has Y as its first letter. For the second letter you find the letter three
places on from L in the alphabet, which gives O as the second letter of the plaintext.
(a) Continue decrypting the code to get the entire original plaintext message.
(b) Describe what you did with Y 2.
(c) Why couldn’t you use any number (e.g. 120) as a key?
5 Decode this message given the key 943.
K X P G Q F O N D M N H K B E L N L N S Y K Z D
Note that you need to work out which order to put the two primes in. Only one will
give you the message.
6 Encrypt your own message using the prime number system. Use only prime numbers
under 100 and keep the message under 25 letters. Give your coded message and key to
someone to decrypt.
e
.au
Research
Find out about the GIMPS (Great Internet Mersenne Prime Search) project and present a
report on how prime numbers are used in e-mail encryption. Discuss whether governments
should ban encryption and make it possible for intelligence agencies to read all e-mails.
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