An Introduction to Mathematical
Cryptography
Solution Manual
Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman
°2008
c by J. Hoffstein, J. Pipher, J.H. Silverman
July 31, 2008
,Chapter 1
An Introduction to
Cryptography
Exercises for Chapter 1
Section. 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 www.math.brown.
edu/~jhs/MathCrypto/CipherWheel.pdf.)
(a) Encrypt the following plaintext using a rotation of 11 clockwise.
“A page of history is worth a volume of logic.”
(b) Decrypt the following message, which was encrypted with a rotation of 7
clockwise.
AOLYLHYLUVZLJYLAZILAALYAOHUAOLZLJYLALZAOHALCLYFIVKFNBLZZLZ
(c) Decrypt the following message, which was encrypted by rotating 1 clock-
wise for the first letter, then 2 clockwise for the second letter, etc.
XJHRFTNZHMZGAHIUETXZJNBWNUTRHEPOMDNBJMAUGORFAOIZOCC
Solution to Exercise 1.1.
apageofhistoryisworthavolumeoflogic
(a)
LALRPZQSTDEZCJTDHZCESLGZWFXPZQWZRTN
This quote is in a court decision of Oliver Wendell Holmes, Jr. (1921).
therearenosecretsbetterthanthesecretsthateverybodyguesses
(b)
AOLYLHYLUVZLJYLAZILAALYAOHUAOLZLJYLAZAOHALCLYFIVKFNBLZZLZ
There are no secrets better than the secrets that everybody
guesses.
This quote is due to George Bernard Shaw, Mrs. Warren’s Profession (1893)
1
,2 Exercises for Chapter 1
whenangrycounttenbeforeyouspeakifveryangryanhundred
(c)
XJHRFTNZHMZGAHIUETXZJNBWNUTRHEPOMDNBJMAUGORFAOIZOCC
When angry, count ten before you speak; if very angry, an hundred.
This quote is due to Thomas Jefferson, A Decalogue of Canons. . . (1825).
1.2. Decrypt each of the following Caesar encryptions by trying the various
possible shifts until you obtain readable text.
(a) LWKLQNWKDWLVKDOOQHYHUVHHDELOOERDUGORYHOBDVDWUHH
(b) UXENRBWXCUXENFQRLQJUCNABFQNWRCJUCNAJCRXWORWMB
(c) BGUTBMBGZTFHNLXMKTIPBMAVAXXLXTEPTRLEXTOXKHHFYHKMAXFHNLX
Solution to Exercise 1.2.
ithinkthatishallneverseeabillboardlovelyasatree
(a)
LWKLQNWKDWLVKDOOQHYHUVHHDELOOERDUGORYHOBDVDWUHH
I think that I shall never see, a billboard lovely as a tree.
This quote is due to Ogden Nash, Many Long Years Ago (1945), Song of the
Open Road.
loveisnotlovewhichalterswhenitalterationfinds
(b)
UXENRBWXCUXENFQRLQJUCNABFQNWRCJUCNAJCRXWORWMB
Love is not love which alters when it alteration finds.
This quote is due to William Shakespeare, Sonnet 116.
inbaitingamousetrapwithcheesealwaysleaveroomforthemouse
(c)
BGUTBMBGZTFHNLXMKTIPBMAVAXXLXTEPTRLEXTOXKHHFYHKMAXFHNLX
In baiting a mousetrap with cheese, always leave room for the
mouse.
This quote is due to H.H. Munro (Saki), The Square Egg (1924).
1.3. For this exercise, use the simple substitution table given in Table 1.11.
(a) Encrypt the plaintext message
The gold is hidden in the garden.
(b) Make a decryption table, that is, make a table in which the ciphertext
alphabet is in order from A to Z and the plaintext alphabet is mixed up.
(c) Use your decryption table from (b) to decrypt the following message.
IBXLX JVXIZ SLLDE VAQLL DEVAU QLB
Solution to Exercise 1.3.
(a)
, Exercises for Chapter 1 3
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 1.1: Simple substitution encryption table for exercise 1.3
t h e g o l d i s h i d d e n i n t h e g a r d e n
I B X F E P A Q L B Q A A X W Q W I B X F S V A X W
Breaking it into five letter blocks gives the ciphertext
IBXFE PAQLB QAAXW QWIBX FSVAX W
(b)
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
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
(c)
t h e s e c r e t p a s s w o r d i s s w o r d f i s h
I B X L X J V X I Z S L L D E V A Q L L D E V A U Q L B
Putting in word breaks gives the plaintext
The secret password is swordfish.
1.4. Each of the following messages has been encrypted using a simple sub-
stitution cipher. Decrypt them. For your convenience, we have given you a
frequency table and a list of the most common bigrams that appear in the
ciphertext. (If you do not want to recopy the ciphertexts by hand, they can
be downloaded or printed from the web site listed in the preface.)
(a) “A Piratical Treasure”
JNRZR BNIGI BJRGZ IZLQR OTDNJ GRIHT USDKR ZZWLG OIBTM NRGJN
IJTZJ LZISJ NRSBL QVRSI ORIQT QDEKJ JNRQW GLOFN IJTZX QLFQL
WBIMJ ITQXT HHTBL KUHQL JZKMM LZRNT OBIMI EURLW BLQZJ GKBJT
QDIQS LWJNR OLGRI EZJGK ZRBGS MJLDG IMNZT OIHRK MOSOT QHIJL
QBRJN IJJNT ZFIZL WIZTO MURZM RBTRZ ZKBNN LFRVR GIZFL KUHIM
MRIGJ LJNRB GKHRT QJRUU RBJLW JNRZI TULGI EZLUK JRUST QZLUK
EURFT JNLKJ JNRXR S
The ciphertext contains 316 letters. Here is a frequency table:
R J I L Z T N Q B G K U M O S H W FE D X V
Freq 33 30 27 25 24 20 19 16 15 15 13 12 12 10 9 8 7 6 5 5 3 2
Cryptography
Solution Manual
Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman
°2008
c by J. Hoffstein, J. Pipher, J.H. Silverman
July 31, 2008
,Chapter 1
An Introduction to
Cryptography
Exercises for Chapter 1
Section. 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 www.math.brown.
edu/~jhs/MathCrypto/CipherWheel.pdf.)
(a) Encrypt the following plaintext using a rotation of 11 clockwise.
“A page of history is worth a volume of logic.”
(b) Decrypt the following message, which was encrypted with a rotation of 7
clockwise.
AOLYLHYLUVZLJYLAZILAALYAOHUAOLZLJYLALZAOHALCLYFIVKFNBLZZLZ
(c) Decrypt the following message, which was encrypted by rotating 1 clock-
wise for the first letter, then 2 clockwise for the second letter, etc.
XJHRFTNZHMZGAHIUETXZJNBWNUTRHEPOMDNBJMAUGORFAOIZOCC
Solution to Exercise 1.1.
apageofhistoryisworthavolumeoflogic
(a)
LALRPZQSTDEZCJTDHZCESLGZWFXPZQWZRTN
This quote is in a court decision of Oliver Wendell Holmes, Jr. (1921).
therearenosecretsbetterthanthesecretsthateverybodyguesses
(b)
AOLYLHYLUVZLJYLAZILAALYAOHUAOLZLJYLAZAOHALCLYFIVKFNBLZZLZ
There are no secrets better than the secrets that everybody
guesses.
This quote is due to George Bernard Shaw, Mrs. Warren’s Profession (1893)
1
,2 Exercises for Chapter 1
whenangrycounttenbeforeyouspeakifveryangryanhundred
(c)
XJHRFTNZHMZGAHIUETXZJNBWNUTRHEPOMDNBJMAUGORFAOIZOCC
When angry, count ten before you speak; if very angry, an hundred.
This quote is due to Thomas Jefferson, A Decalogue of Canons. . . (1825).
1.2. Decrypt each of the following Caesar encryptions by trying the various
possible shifts until you obtain readable text.
(a) LWKLQNWKDWLVKDOOQHYHUVHHDELOOERDUGORYHOBDVDWUHH
(b) UXENRBWXCUXENFQRLQJUCNABFQNWRCJUCNAJCRXWORWMB
(c) BGUTBMBGZTFHNLXMKTIPBMAVAXXLXTEPTRLEXTOXKHHFYHKMAXFHNLX
Solution to Exercise 1.2.
ithinkthatishallneverseeabillboardlovelyasatree
(a)
LWKLQNWKDWLVKDOOQHYHUVHHDELOOERDUGORYHOBDVDWUHH
I think that I shall never see, a billboard lovely as a tree.
This quote is due to Ogden Nash, Many Long Years Ago (1945), Song of the
Open Road.
loveisnotlovewhichalterswhenitalterationfinds
(b)
UXENRBWXCUXENFQRLQJUCNABFQNWRCJUCNAJCRXWORWMB
Love is not love which alters when it alteration finds.
This quote is due to William Shakespeare, Sonnet 116.
inbaitingamousetrapwithcheesealwaysleaveroomforthemouse
(c)
BGUTBMBGZTFHNLXMKTIPBMAVAXXLXTEPTRLEXTOXKHHFYHKMAXFHNLX
In baiting a mousetrap with cheese, always leave room for the
mouse.
This quote is due to H.H. Munro (Saki), The Square Egg (1924).
1.3. For this exercise, use the simple substitution table given in Table 1.11.
(a) Encrypt the plaintext message
The gold is hidden in the garden.
(b) Make a decryption table, that is, make a table in which the ciphertext
alphabet is in order from A to Z and the plaintext alphabet is mixed up.
(c) Use your decryption table from (b) to decrypt the following message.
IBXLX JVXIZ SLLDE VAQLL DEVAU QLB
Solution to Exercise 1.3.
(a)
, Exercises for Chapter 1 3
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 1.1: Simple substitution encryption table for exercise 1.3
t h e g o l d i s h i d d e n i n t h e g a r d e n
I B X F E P A Q L B Q A A X W Q W I B X F S V A X W
Breaking it into five letter blocks gives the ciphertext
IBXFE PAQLB QAAXW QWIBX FSVAX W
(b)
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
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
(c)
t h e s e c r e t p a s s w o r d i s s w o r d f i s h
I B X L X J V X I Z S L L D E V A Q L L D E V A U Q L B
Putting in word breaks gives the plaintext
The secret password is swordfish.
1.4. Each of the following messages has been encrypted using a simple sub-
stitution cipher. Decrypt them. For your convenience, we have given you a
frequency table and a list of the most common bigrams that appear in the
ciphertext. (If you do not want to recopy the ciphertexts by hand, they can
be downloaded or printed from the web site listed in the preface.)
(a) “A Piratical Treasure”
JNRZR BNIGI BJRGZ IZLQR OTDNJ GRIHT USDKR ZZWLG OIBTM NRGJN
IJTZJ LZISJ NRSBL QVRSI ORIQT QDEKJ JNRQW GLOFN IJTZX QLFQL
WBIMJ ITQXT HHTBL KUHQL JZKMM LZRNT OBIMI EURLW BLQZJ GKBJT
QDIQS LWJNR OLGRI EZJGK ZRBGS MJLDG IMNZT OIHRK MOSOT QHIJL
QBRJN IJJNT ZFIZL WIZTO MURZM RBTRZ ZKBNN LFRVR GIZFL KUHIM
MRIGJ LJNRB GKHRT QJRUU RBJLW JNRZI TULGI EZLUK JRUST QZLUK
EURFT JNLKJ JNRXR S
The ciphertext contains 316 letters. Here is a frequency table:
R J I L Z T N Q B G K U M O S H W FE D X V
Freq 33 30 27 25 24 20 19 16 15 15 13 12 12 10 9 8 7 6 5 5 3 2
