Cryptography Homework #1 Solutions
1) THE BRITISH ARE COMING THE BRITISH ARE COMING (Shift =
11)
2) THIS CIPHER TESTS WHETHER OR NOT SPACES MATTER (Shift=4)
3) FOR THOUSANDS OF YEARS KINGS QUEENS AND GENERALS HAVE
RELIED ON EFFICIENT COMMUNICATION IN ORDER TO GOVERN THEIR
COUNTRIES AND COMMAND THEIR ARMIES (a=15, b=9, for
decryption, the encryption keys were a=7, b=15)
4) I HAVE PLACED A PRIZE IN THE BUILDING IN WHICH MY OFFICE
IS AND GO NEAR ROOM TWO FOUR SIX THERE YOU WILL FIND A FIRE
DEPARTMENT VALVE LOOK INSIDE OF IT AND YOU WILL SEE YOUR
PRIZE I AM HIDING THE PRIZE TUESDAY AFTERNOON GOOD LUCK (a
= 9, b = 17 are encryption keys.)
5) study this paragraph and all things in it what is
vitally wrong with it actually nothing in it is wrong but
you must admit that it is most unusual do not just zip
through it quickly but study it scrupulously with luck you
should spot what is so particular about it and all words
found in it can you say what it is tax your brains and try
again do not miss a word or a symbol it is not all that
difficult (a=5, b=12, for decryption, the encryption keys
were a=21, b=8)
6) Our encryption function is as follows:
f(x) = (7x + 12) mod 26
To find its inverse, replace f(x) with x and x with f-1(x):
x = (7f-1(x) + 12) mod 26
x – 12 = 7f-1(x) mod 26
Since the modular inverse of 7 mod 26 is 15, our next step is as follows:
15(x – 12) = 15(7)f-1(x) mod 26
15x – 180 = f-1(x) mod 26
f-1(x) = (15x + 2) mod 26
7) We need to determine how many integers in [1, 39] are relatively prime to 40. 40’s
distinct prime factors are 2 and 5. Thus, this list is as follows: 1, 3, 7, 9, 11, 13, 17, 19,
21, 23, 27, 29, 31, 33, 37 and 39. There are 16 such values. It follows that the total
number of possible keys for an affine cipher is 16 x 40 = 640.