Recap
Definition (Encryption: Caesar Cipher)
+3=⇥
mod 26
Recap
Definition (Encryption: Caesar Cipher)
+3=⇥
mod 26
Definition (Encryption: Shift Cipher)
+d =⇥
mod 26, d a number
Recap
Definition (Encryption: Caesar Cipher)
+3=⇥
mod 26
Definition (Encryption: Shift Cipher)
+d =⇥
mod 26, d a number
Definition (Decryption: Caeser Cipher)
+ 23 = ⇥
mod 26
Recap
Definition (Encryption: Caesar Cipher)
+3=⇥
mod 26
Definition (Encryption: Shift Cipher)
+d =⇥
mod 26, d a number
Definition (Decryption: Caeser Cipher)
+ 23 = ⇥
mod 26
Definition (Decryption: Shift Cipher)
⇥ + d mod 26 =
, d is the additive inverse of d
Code Summary: Caesar Cipher
We will talk about several di↵erent types of codes during the next
few weeks and it will be good to keep a summary for each.
Code Summary: Caesar Cipher
We will talk about several di↵erent types of codes during the next
few weeks and it will be good to keep a summary for each.
• Encryption/Decryption Key
Code Summary: Caesar Cipher
We will talk about several di↵erent types of codes during the next
few weeks and it will be good to keep a summary for each.
• Encryption/Decryption Key
• Key Secrecy is the idea of how secret the decryption key must
be. There are codes where anyone can have access to the key!
Code Summary: Caesar Cipher
We will talk about several di↵erent types of codes during the next
few weeks and it will be good to keep a summary for each.
• Encryption/Decryption Key
• Key Secrecy is the idea of how secret the decryption key must
be. There are codes where anyone can have access to the key!
• Letter Frequency, or how much a cipher changes the nature of
how often letters appear, will become increasingly important.
Code Summary: Caesar Cipher
We will talk about several di↵erent types of codes during the next
few weeks and it will be good to keep a summary for each.
• Encryption/Decryption Key
• Key Secrecy is the idea of how secret the decryption key must
be. There are codes where anyone can have access to the key!
• Letter Frequency, or how much a cipher changes the nature of
how often letters appear, will become increasingly important.
Cipher
Caesar
Shift
Encrypt
Key(s)
3
d
Decrypt
Key(s)
23
d
Key
Secrecy
Private
Private
Letter
Frequency
Normal
Normal
Related Idea: Frequency Analysis
Anyone who has watched Wheel of Fortune or played Scrabble
knows that the English language uses some letters more frequently
than others.
E
T
A
NO
I
CD
B
H
F G
R S
L
M
J K
U
P
Q
V
W
Some codes do not hide the natural frequency of letters.
Y
X
Z
Related Idea: Frequency Analysis
The Caesar Cipher disguises letters, but does not disguise the
natural frequency of letters!
H
W
D
F G
B
A
C
QR
L
E
K
I J
UV
O
P
MN
X
S
T
Y
Z
The most frequent symbol used in the ciphertext will correspond to
the letter “E” in the plaintext. For the Caesar Cipher, this
corresponds to the numeric 5 + 3 and is the ciphertext letter “H”.
Recap
Recall that by simplifying the statement a mod n we mean to find
a number b between 0 and n 1 such that a = b mod n.
Example
1. Simplify 14 mod 3.
2. Simplify 78 mod 25.
3. Simplify 29 mod 26.
Related Idea: Additive Inverse
Definition (Related Idea: Additive Inverse)
The additive inverse for a mod n is a value a so that
a + a = 0 mod n
Related Idea: Additive Inverse
Definition (Related Idea: Additive Inverse)
The additive inverse for a mod n is a value a so that
a + a = 0 mod n
From working with codes, we can understand that using the
additive inverse d “really” works by moving all letters to the left d
places.
Related Idea: Additive Inverse
Definition (Related Idea: Additive Inverse)
The additive inverse for a mod n is a value a so that
a + a = 0 mod n
From working with codes, we can understand that using the
additive inverse d “really” works by moving all letters to the left d
places.
Example
1. Find the additive inverse of 8 mod 10.
Related Idea: Additive Inverse
Definition (Related Idea: Additive Inverse)
The additive inverse for a mod n is a value a so that
a + a = 0 mod n
From working with codes, we can understand that using the
additive inverse d “really” works by moving all letters to the left d
places.
Example
1. Find the additive inverse of 8 mod 10.
2. Find the additive inverse of 5 mod 12.
Related Idea: Additive Inverse
Definition (Related Idea: Additive Inverse)
The additive inverse for a mod n is a value a so that
a + a = 0 mod n
From working with codes, we can understand that using the
additive inverse d “really” works by moving all letters to the left d
places.
Example
1. Find the additive inverse of 8 mod 10.
2. Find the additive inverse of 5 mod 12.
3. Find the additive inverse of 7 mod 14.
Related Idea: Simplifying Negative Mods, Method 1
Recall that by simplifying the statement a mod n we mean to find
a number b between 0 and n 1 such that a = b mod n.
Related Idea: Simplifying Negative Mods, Method 1
Recall that by simplifying the statement a mod n we mean to find
a number b between 0 and n 1 such that a = b mod n.
Simplifying the statement “ a mod n” for a > 0 means do the
following:
1. First simplify a mod n, say a = b mod n.
Related Idea: Simplifying Negative Mods, Method 1
Recall that by simplifying the statement a mod n we mean to find
a number b between 0 and n 1 such that a = b mod n.
Simplifying the statement “ a mod n” for a > 0 means do the
following:
1. First simplify a mod n, say a = b mod n.
2. Determine the additive inverse of b mod n.
Related Idea: Simplifying Negative Mods, Method 1
Recall that by simplifying the statement a mod n we mean to find
a number b between 0 and n 1 such that a = b mod n.
Simplifying the statement “ a mod n” for a > 0 means do the
following:
1. First simplify a mod n, say a = b mod n.
2. Determine the additive inverse of b mod n.
Example (Additive Inverses and Negatives)
Simplify
4 mod 10 .
Related Idea: Simplifying Negative Mods, Method 1
Recall that by simplifying the statement a mod n we mean to find
a number b between 0 and n 1 such that a = b mod n.
Simplifying the statement “ a mod n” for a > 0 means do the
following:
1. First simplify a mod n, say a = b mod n.
2. Determine the additive inverse of b mod n.
Example (Additive Inverses and Negatives)
Simplify
4 mod 10 .
Simplify
20 mod 16 .
Related Idea: Simplifying Negative Mods, Method 1
Recall that by simplifying the statement a mod n we mean to find
a number b between 0 and n 1 such that a = b mod n.
Simplifying the statement “ a mod n” for a > 0 means do the
following:
1. First simplify a mod n, say a = b mod n.
2. Determine the additive inverse of b mod n.
Example (Additive Inverses and Negatives)
Simplify
4 mod 10 .
Simplify
20 mod 16 .
Simplify
14 mod 5 .
SIDE A
Introduction to Vigenere Ciphers
Example
Enemy agents have started to make their codes more
sophisticated. They now use multiple shifts at once! You intercept
a message and learn that the shifts being used correspond to
d1 = 4, d2 = 15, d3 = 7.
How would you encrypt the plaintext “ATE”?
How would you encrypt the plaintext “TEA”?
Encryption Method: Vigenère Cipher
Definition
An English Language Vigenère Cipher uses a di↵erent shift for
each letter, depending on the position of the letter in the message.
Encryption uses the rule
1
+d1 mod 26 = ⇥1
2
+d2 mod 26 = ⇥2
3
+d3 mod 26 = ⇥3
..
.
where 1 is the first plaintext letter and d1 is the first shift,
the second plaintext letter and d2 is the second shift, etc...
2
is
Example
Example
Suppose you use a Vigenère Cipher with the following shifts.
d1 = 4, d2 = 15, d3 = 7.
Example
Example
Suppose you use a Vigenère Cipher with the following shifts.
d1 = 4, d2 = 15, d3 = 7.
Encrypt the plaintext “ATE”.
Example
Example
Suppose you use a Vigenère Cipher with the following shifts.
d1 = 4, d2 = 15, d3 = 7.
Encrypt the plaintext “ATE”.
Encrypt the plaintext “TEA”.
Example
Example
Suppose you use a Vigenère Cipher with the following shifts.
d1 = 4, d2 = 15, d3 = 7.
Encrypt the plaintext “ATE”.
Encrypt the plaintext “TEA”.
Encrypt these two words using the Caeser Cipher and compare
the encrypted words to the ones you got above.
Related Idea: Keyword
Definition
We use a Keyword to represent all of the di↵erent shifts (and the
order) to be used with a Vigenère Cipher.
Related Idea: Keyword
Definition
We use a Keyword to represent all of the di↵erent shifts (and the
order) to be used with a Vigenère Cipher.
Example
The keyword ENEMY gives the following shifts:
d1 = 5
d2 = 14
d3 = 5
d4 = 13
d5 = 25
So the first letter in our message will get shifted 5 places to the
right, the second letter will get shifted 14 places to the right, and
so on.
One more example!
Example
Encode the word DUMPLING using the keyword ARENA.
One more example!
Example
Encode the word DUMPLING using the keyword ARENA.
A=1
R=18
E=5
N=14
A=1
One more example!
Example
Encode the word DUMPLING using the keyword ARENA.
A=1
R=18
E=5
N=14
DUMPLING 7! EMRDMJFL
A=1
One more example!
Example
Encode the word DUMPLING using the keyword ARENA.
A=1
R=18
E=5
N=14
DUMPLING 7! EMRDMJFL
What if we used the keyword GOB?
A=1
One more example!
Example
Encode the word DUMPLING using the keyword ARENA.
A=1
R=18
E=5
N=14
DUMPLING 7! EMRDMJFL
What if we used the keyword GOB?
G=7
O=15
B=2
A=1
One more example!
Example
Encode the word DUMPLING using the keyword ARENA.
A=1
R=18
E=5
N=14
DUMPLING 7! EMRDMJFL
What if we used the keyword GOB?
G=7
O=15
B=2
DUMPLING 7! KJOWAKUV
A=1
SIDE B (Part 1)
Decryption Method: Vigenère Cipher
Definition
A Vigenère Cipher can be decrypted as follows:
(i) Identify d1 ,d2 , d3 , etc . . . corresponding to the letters of the
Keyword.
(ii) Find the additive inverse d 1 to d1 . Next find the
additive inverse d 2 to d2 . Continue for all values dk .
Together, the numbers d 1 , d 2 , d 3 , . . . are called the Decryption
Sequence.
To decrypt a Vigenère Cipher, we use the rule
⇥k + d k mod 26 =
k.
Example
Example ( Example )
The enemy agent starts using a new keyword. You MUST break
this new code! You figure out the shifts being used are
d1 = 1, d2 = 16, d3 = 16 d4 = 12, d5 = 5.
What Keyword is being used for this Vigenère Cipher?
Example
Example ( Example )
The enemy agent starts using a new keyword. You MUST break
this new code! You figure out the shifts being used are
d1 = 1, d2 = 16, d3 = 16 d4 = 12, d5 = 5.
What Keyword is being used for this Vigenère Cipher?
What is the decryption sequence?
Example
Example ( Example )
The enemy agent starts using a new keyword. You MUST break
this new code! You figure out the shifts being used are
d1 = 1, d2 = 16, d3 = 16 d4 = 12, d5 = 5.
What Keyword is being used for this Vigenère Cipher?
What is the decryption sequence?
Decrypt the ciphertext message “FDUYD”.
Related Idea: Advantages of the Vigenère Cipher
The Vigenère Cipher is a HUGE improvement over Shift Ciphers
because any single letter can be encrypted in many di↵erent ways.
Related Idea: Advantages of the Vigenère Cipher
The Vigenère Cipher is a HUGE improvement over Shift Ciphers
because any single letter can be encrypted in many di↵erent ways.
Theorem (Counting Possibilities for the Vigenère Cipher)
The number of di↵erent ways a letter can be encrypted using the
Vigenère Cipher is at most the length of the keyword being used.
Related Idea: Advantages of the Vigenère Cipher
The Vigenère Cipher is a HUGE improvement over Shift Ciphers
because any single letter can be encrypted in many di↵erent ways.
Theorem (Counting Possibilities for the Vigenère Cipher)
The number of di↵erent ways a letter can be encrypted using the
Vigenère Cipher is at most the length of the keyword being used. If
the keyword has no repeats in letters then this number is exactly
the length of the keyword.
Related Idea: Advantages of the Vigenère Cipher
The Vigenère Cipher is a HUGE improvement over Shift Ciphers
because any single letter can be encrypted in many di↵erent ways.
Theorem (Counting Possibilities for the Vigenère Cipher)
The number of di↵erent ways a letter can be encrypted using the
Vigenère Cipher is at most the length of the keyword being used. If
the keyword has no repeats in letters then this number is exactly
the length of the keyword.
Example
For a Vigenère Cipher that uses the keyword “TWIN”, the letter
“E” could be encrypted using “T”, “W”, “I”, or “N”. This means
that “E” could be encrypted as
Plaintext Numeric
Keyword Numeric
Ciphertext ⇥
5
20
Y
5
23
B
5
9
N
5
14
S