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Introduction to Cryptography
UNIK 4220 (Fall 2017)
Mandatory assignment
Leif Nilsen
October 4, 2017
General
These exercises constitute a mandatory part of the course UNIK 4220 – Introduction to Cryptography. This means that an individual or pairwise solution proposal must be submitted and approved
before you are allowed access to the final exam in December. Your solution should be in electronic
format (pdf, ps, Word) and must be uploaded to the ”Innleveringsmappe” catalog in Fronter before
24:00 on Sunday 22nd October. You are free to use any electronic tools you prefer, but all exercises
are in principle solvable using manual methods. Ensure that your submission includes your name
or both names in case of a pairwise solution.
Explaining your principles and solution strategies are more important than conducting lengthy
calculations. The grading will be APPROVED/NOT APPROVED and 30% score will be required
for approval. Ensure that none of the 5 exercises are completely blank.
Best wishes!
Exercise 1
Oscar has obtained the following ciphertext from Alice. He knows that the corresponding plaintext
is English text without space characters as word separators and that Alice and Bob use the affine
cipher to secure their communication.
a) Explain the steps Oscar needs to perform in order to recover the plaintext.
b) Find the encryption key.
c) Compute the decryption key.
d) Find the plaintext.
WXXWSCGJKXVOWZZGJOSGFVOBSWJHODUJOHAKIOMMGJKX
VOYUMXZBOQIOJXSVWBWSXOBMIMGJKZBOQIOJSAWJWNAM
GMXVOKIOMMGJKEGNNNOWDXUWNGJOWBOQIWXGUJMAMXOY
EGXVXEUIJCJUEJWJDXEUOQIWXGUJMXVOMUNIXGUJEGNN
FBUTGDOWSWJDGDWXOCOAXVOCOAEGNNXVOJHOTOBGZGOD
HADOSBAFXGUJUZXVOSGFVOBXOPX
1
Exercise 2
In this exercise you shall use a standard 3-rotors service Enigma. Use the following configuration
Rotor order: 1 – 2 – 3
Ring order: 01 – 01 – 01 (A – A – A)
Starting position: 01 – 04 – 21 (A – D – U)
Stecker: None
a) Determine the permutation that Enigma defines in this position. The permutation shall be
given both in table form and cycle form.
b) Encrypt three characters while you watch the rotor windows. What do you see? Explain
what is happening.
c) What is the effect of this phenomena to the period of the Enigma? How many letters must
be encrypted before Enigma starts to repeat the substitution alphabets?
Exercise 3
Alice and Bob have agreed to encrypt their communication using a stream cipher. Their system is
based on a LFSR of length 8. Plaintext and ciphertext are given by the 26 letters in the English
language augmented with “Space”, period, comma, question mark, dash and colon. The characters
are encoded as 5 binary digits as follows: A → “00000”, B → “00001”, …, Z → “11001”, and Space
→ “11010”, “.” → “11011” , “,” → “11100”, ? → “11101”, – → “11110” and : → “11111”.
a) Alice and Bob have agreed to use their key as the initial value of a LFSR defined by the
characteristic polynomial f (x) = 1 + x + x5 + x6 + x8 . One day Oscar sees the ciphertext
“KIHJFRTXRNQLW,UDXJRAWUQJLA L T S”. Explain how Oscar can decrypt this message when he knows f (x) and has learned that Alice always starts her messages with “HELLO”.
b) Find the corresponding plaintext.
c) Alice and Bob understand that this system is unsecure, and decide to use a secret characteristic polynomial. This time Oscar sees the ciphertext
“WQJCA:X.CW:LYDQ:SU-GKNUQDH”. Find the corresponding plaintext knowing that Alice still starts her messages with “HELLO”.
Exercise 4
We define the finite field F = Z2 [x]/(m(x)) using the polynomial m(x) = x8 + x4 + x3 + x + 1. The
elements of F are represented as octets XY in hex notation.
a) Let a = 06 and b = 2E be two elements in F. Compute a + b, ab and b2 when all calculations
take place in F.
b) Suppose two polynomials c(x) and d(x) with coefficients from F are given by :
c(x) = 03×3 + 01×2 + 01x + 02
2
d(x) = 0Bx3 + 0Dx2 + 09x + 0E
Show that c(x)d(x) = 01(mod(x4 + 01)). The polynomial c(x) defines the MixColumns
mapping in Rijndael, and d(x) defines the inverse mapping.
Exercise 5
Alice and Bob have agreed that Alice every morning shall send an encrypted messages that consists
of just 1 bit, i.e. the plaintext shall be 0 or 1. For their usages they have an implementation of
standard AES in encryption and decryption direction, and they have by a secure operation agreed
on a common symmetric key kAB .
a) Explain how you will use the AES-algorithm to design a solution to this problem. You are free
to choose mode of operation and to suggest appropriate protocols that are needed. It is not
required that the plaintext and ciphertext shall have the same length. Justify all assumptions
and design choices you make.
b) Explain why you consider your design to provide a “secure” solution in this situation.
3
Introduction to Cryptography
UNIK 4220 (Fall 2017)
Mandatory assignment
Leif Nilsen
October 4, 2017
General
These exercises constitute a mandatory part of the course UNIK 4220 – Introduction to Cryptography. This means that an individual or pairwise solution proposal must be submitted and approved
before you are allowed access to the final exam in December. Your solution should be in electronic
format (pdf, ps, Word) and must be uploaded to the ”Innleveringsmappe” catalog in Fronter before
24:00 on Sunday 22nd October. You are free to use any electronic tools you prefer, but all exercises
are in principle solvable using manual methods. Ensure that your submission includes your name
or both names in case of a pairwise solution.
Explaining your principles and solution strategies are more important than conducting lengthy
calculations. The grading will be APPROVED/NOT APPROVED and 30% score will be required
for approval. Ensure that none of the 5 exercises are completely blank.
Best wishes!
Exercise 1
Oscar has obtained the following ciphertext from Alice. He knows that the corresponding plaintext
is English text without space characters as word separators and that Alice and Bob use the affine
cipher to secure their communication.
a) Explain the steps Oscar needs to perform in order to recover the plaintext.
b) Find the encryption key.
c) Compute the decryption key.
d) Find the plaintext.
WXXWSCGJKXVOWZZGJOSGFVOBSWJHODUJOHAKIOMMGJKX
VOYUMXZBOQIOJXSVWBWSXOBMIMGJKZBOQIOJSAWJWNAM
GMXVOKIOMMGJKEGNNNOWDXUWNGJOWBOQIWXGUJMAMXOY
EGXVXEUIJCJUEJWJDXEUOQIWXGUJMXVOMUNIXGUJEGNN
FBUTGDOWSWJDGDWXOCOAXVOCOAEGNNXVOJHOTOBGZGOD
HADOSBAFXGUJUZXVOSGFVOBXOPX
1
Exercise 2
In this exercise you shall use a standard 3-rotors service Enigma. Use the following configuration
Rotor order: 1 – 2 – 3
Ring order: 01 – 01 – 01 (A – A – A)
Starting position: 01 – 04 – 21 (A – D – U)
Stecker: None
a) Determine the permutation that Enigma defines in this position. The permutation shall be
given both in table form and cycle form.
b) Encrypt three characters while you watch the rotor windows. What do you see? Explain
what is happening.
c) What is the effect of this phenomena to the period of the Enigma? How many letters must
be encrypted before Enigma starts to repeat the substitution alphabets?
Exercise 3
Alice and Bob have agreed to encrypt their communication using a stream cipher. Their system is
based on a LFSR of length 8. Plaintext and ciphertext are given by the 26 letters in the English
language augmented with “Space”, period, comma, question mark, dash and colon. The characters
are encoded as 5 binary digits as follows: A → “00000”, B → “00001”, …, Z → “11001”, and Space
→ “11010”, “.” → “11011” , “,” → “11100”, ? → “11101”, – → “11110” and : → “11111”.
a) Alice and Bob have agreed to use their key as the initial value of a LFSR defined by the
characteristic polynomial f (x) = 1 + x + x5 + x6 + x8 . One day Oscar sees the ciphertext
“KIHJFRTXRNQLW,UDXJRAWUQJLA L T S”. Explain how Oscar can decrypt this message when he knows f (x) and has learned that Alice always starts her messages with “HELLO”.
b) Find the corresponding plaintext.
c) Alice and Bob understand that this system is unsecure, and decide to use a secret characteristic polynomial. This time Oscar sees the ciphertext
“WQJCA:X.CW:LYDQ:SU-GKNUQDH”. Find the corresponding plaintext knowing that Alice still starts her messages with “HELLO”.
Exercise 4
We define the finite field F = Z2 [x]/(m(x)) using the polynomial m(x) = x8 + x4 + x3 + x + 1. The
elements of F are represented as octets XY in hex notation.
a) Let a = 06 and b = 2E be two elements in F. Compute a + b, ab and b2 when all calculations
take place in F.
b) Suppose two polynomials c(x) and d(x) with coefficients from F are given by :
c(x) = 03×3 + 01×2 + 01x + 02
2
d(x) = 0Bx3 + 0Dx2 + 09x + 0E
Show that c(x)d(x) = 01(mod(x4 + 01)). The polynomial c(x) defines the MixColumns
mapping in Rijndael, and d(x) defines the inverse mapping.
Exercise 5
Alice and Bob have agreed that Alice every morning shall send an encrypted messages that consists
of just 1 bit, i.e. the plaintext shall be 0 or 1. For their usages they have an implementation of
standard AES in encryption and decryption direction, and they have by a secure operation agreed
on a common symmetric key kAB .
a) Explain how you will use the AES-algorithm to design a solution to this problem. You are free
to choose mode of operation and to suggest appropriate protocols that are needed. It is not
required that the plaintext and ciphertext shall have the same length. Justify all assumptions
and design choices you make.
b) Explain why you consider your design to provide a “secure” solution in this situation.
3
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