An Introduction to Mathematical Cryptography Jeffrey Hoffstein, Joseph H. Silverman, Jill Pipher

Proposed Solutions to the questions in the book An Introduction to Mathematical Cryptography Jeffrey Hoffstein, Joseph H. Silverman, Jill Pipher

Chapter 1 : An Introduction to Cryptography

Section 1.1. Simple substitution ciphers:

1.1. Build a cipher wheel as illustrated in Figure 1.1, but with an inner wheel that rotates, and use it to complete the following tasks. (For your convenience, there is a cipher wheel that you can print and cut out at MathCrypto/CipherWheel )

1. Encrypt the following plaintext using a rotation of 11 clockwise.

   “A page of history is worth a volume of logic.”

a -L b-M c-N d-O e-P f-Q g-R h-S i-T j-U k-V l-W m-X n-Y o-Z p-A q-B r-C s-D t-E u-F v-G w-H x-I y-J z-K

MALRPZQSTDEZCJTDHZCESLGZWFXPZQWZRTN

2. Decrypt the following message, which was encrypted with a rotation of 7 clockwise.

   AOLYLHYLUVZLJYLAZILAALYAOHUAOLZLJYLALZAOHALCLYFIVKFNBLZZLZ

a -H b-I c-J d-K e-L f-M g-N h-O i-P j-Q k-R l-S m-T n-U o-V p-W q-X r-Y s-Z t-A u-B v-C w-D x-E y-F z-G

"There are no secrets better than the secretes that everybody guesses"

3. Decrypt the following message, which was encrypted by rotating 1 clockwise for the first letter, then 2 clockwise for the second letter, etc.

   XJHRFTNZHMZGAHIUETXZJNBWNUTRHEPOMDNBJMAUGORFAOIZOCC

A– a B– b C– c D– d E– e F – f G – g H – h I – i J – j K – k L – l M – m N– n O– o P – p Q – q R – r S – s T – t U – u V – v W – w X– x Y – y Z – z

"When angry count ten before you speak if very angry an hundred"

1.2. Decrypt each of the following Caesar encryptions by trying the various possible shifts until you obtain readable text.

1. LWKLQNWKDWLVKDOOQHYHUVHHDELOOERDUGORYHOBDVDWUHH
2. UXENRBWXCUXENFQRLQJUCNABFQNWRCJUCNAJCRXWORWMB
3. BGUTBMBGZTFHNLXMKTIPBMAVAXXLXTEPTRLEXTOXKHHFYHKMAXFHNLX

1.3. For this exercise, use the simple substitution table given in Table 1.11.

a

b

c

d

e

f

g

h

i

j

k

l

m

n

o

p

q

r

s

t

u

v

w

x

y

z

S

C

J

A

X

U

F

B

Q

K

T

P

R

W

E

Z

H

V

L

I

G

Y

D

N

M

O

Table 1.11: Simple substitution encryption table for exercise 1.3

1. Encrypt the plaintext message

   The gold is hidden in the garden.

IBX FESA QL BQAAXR QW IBX FSVAXW
2. Make a decryption table, that is, make a table in which the cipher text alphabetis in order from A to Z and the plaintext alphabet is mixed up.

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

d

h

b

w

o

g

u

q

t

c

j

s

y

x

z

l

i

m

a

k

f

r

n

e

v

p

3. Use your decryption table from (b) to decrypt the following message.

   IBXLX JVXIZ SLLDE VAQLL DEVAU QLB

"The secret password is swordfish"

1.5. Suppose that you have an alphabet of 26 letters.

1. How many possible simple substitution ciphers are there?
The total number of ways to assign the 26 plaintext letters to the 26 ciphertext letters, using each ciphertext letter only once, is

26 * 25 * 24 · · · 4 * 3 * 2 * 1 = 26! = 403291461126605635584000000.