GT​ CS 6035: Introduction to Information Security
Project ​3​ :
All Things Cryptography
Summer 2020
The goals of this project :

Students will advance their knowledge of cryptography and hashing by working through example
exercises and then trying to exploit vulnerable systems.
Preface :
BEFORE STARTING, MAKE SURE YOU ARE USING PYTHON
VERSION 3.7.x OR LOWER​. VERSION 3.8 INCLUDES SOME
FUNCTIONALITY THAT MAY NOT BE COMPATIBLE WITH THE
VERSION 3.6.9. TO CHECK YOUR VERSION OF PYTHON, OPEN
A COMMAND PROMPT AND RUN THE COMMAND:
python --version
FOR THE ESTABLISHED ALGORITHMS THAT YOU MAY NEED
TO USE, YOU ARE ALLOWED TO IMPLEMENT PSEUDOCODE
WITH PROPER CITATION. What is Pseudocode? -->
https://en.wikipedia.org/wiki/Pseudocode
HOWEVER, UNDER NO CIRCUMSTANCES SHOULD YOU
COPY/PASTE CODE INTO THE PROJECT. DOING SO IS AN
HONOR CODE VIOLATION (NOT TO MENTION A REAL WORLD
SECURITY CONCERN).

GT​ CS 6035: Introduction to Information Security
Intro :

RSA is one of the most widely-used public key cryptosystems in the world. It’s composed of three
algorithms: key generation (Gen), encryption (Enc), and decryption (Dec). In RSA, the public key is
a pair of integers , and the private key is an integer .e, N )( d

The key pair is generated by the following steps:

1. Choose two distinct big prime numbers with the same bit size, say and .p q

2. Let , and . p ∗ qN = (N ) (p − 1) ∗ (q − 1)φ =

3. Pick up an integer , such that and .e e φ(N )1 < < cd(e, φ(N )) 1g =

4. Get the modular inverse of .d ≡ e mod φ(N ) (i.e., d ∗ e ≡ 1 mod φ(N ))e : −1

5. Return as public key, and d as private key.N , e)(

Enc -​ To encrypt integer m with public key , the cipher integer .N , e)( mod Nc ≡ m e
Dec​ - To decrypt cipher integer c with private key d, the plain integer . mod Nm ≡ c d

Task 1 – Warm-up, Get Familiar with RSA - (​5​ points)

The goal of this task is to get you familiar with RSA. You are given an RSA key pair and ,N , e)( d
and a unique encrypted message . You are required to get the decrypted message .c m

TODO:​ In the provided ​project_3.py​ file, implement the stub method ​task_1​. ​Hint:​ Don’t
overthink it, this can be done with a single Python command…

def​ ​task​_​1​(self, n_str: str, d_str: str, c_str: str):
# TODO: Implement this method for Task 1
n = int(n_str, 16)
d = int(d_str, 16)
c = int(c_str, 16)
m = ​0

​return​ hex(m).rstrip(​'L'​)

Task 2 – Warm-up, Get Familiar with Hashes (​10​ points)

By now we’ve learned that hashes are one-way functions. Because of this unique feature,
passwords are often stored as hashes in order to protect them from prying eyes. Even if a hacker
infiltrated our state-of-the-art Georgia Tech security systems, he or she would not be able to derive
the plaintext passwords from the hashes. But what if we made the critical mistake of using a

GT​ CS 6035: Introduction to Information Security
common password? ​How safe would we be?

Let’s find out...

You are given a list of some of the most commonly-used passwords on the Internet. You are also
given the ​SHA256​ hash of a password randomly selected from this list. Your job is to discover the

The complete list of common passwords is pre-loaded for you in ​project_3.py​.

TODO:​ In the provided ​project_3.py​ file, implement the stub method ​task_2​.

# TODO: Implement this method for Task 2

# This is how you get the SHA-256 hash:

Reflection
In a maximum of 200 words, address the following prompt:

Knowing that a lot of people like to use these common passwords, make one suggestion for how
you could implement improved password security.
Task 3 – Kernelcoin (​15​ points)

Background: A blockchain is a distributed, immutable ledger that derives its security, in part, from a
chain of cryptographic hash values. For more detail, please read Section II of Hassan et al.,
Blockchain and the Future of the Internet: A Comprehensive Review, arXiv:1904.00733v1 (23 Feb.
2019), available online at: ​https://arxiv.org/pdf/1904.00733.pdf​.

Today is your lucky day! You’ve discovered a brand new cryptocurrency called kernelcoin (symbol:
RTI). There are rumors that Costco will soon announce kernelcoin as the preferred payment
method in their warehouse stores. This news is sure to send the price of kernelcoin to the moon,
and kernelcoin holders to the nearest Lamborghini dealership.

You plan to start mining kernelcoin so that you can earn even more. In order to do so, you need to
create a valid block to append to the previous block. A valid block contains the lowest nonce value
that, when concatenated with the transaction string, and the hash of the previous block (in that
order, i.e. nonce + transaction string + previous block hash), will produce a SHA256 hash with two
leading zeros (the proof-of-work for this particular blockchain). Transaction strings have the syntax
“UserID1:UserID2:X”, indicating that UserID1has transferred X kernelcoin to UserID2. You are given
all of these values, and your goal is to find the lowest possible nonce value for the resulting block.

GT​ CS 6035: Introduction to Information Security
TODO:​ In the provided ​project_3.py​ file, implement the method ​task_3​.

def​ ​task​_​3​(self, user_id_1: str, user_id_2: str, amount: int, prev_block_hash:
str):
# TODO: Implement this method for Task 3
nonce = 0

​return​ nonce

Reflection
In a maximum of 200 words, address the following prompt:

The kernelcoin blockchain uses a proof-of-work scheme as a consensus mechanism (i.e., finding a
hash with a certain number of leading zeros). Name an alternative consensus mechanism and list
its strengths and weaknesses compared to proof-of-work.
Task 4 – Attack A Small Key Space (​15​ points)

The algorithm you search for is dirt simple which makes it hard for attackers to traverse the entire
key space with limited resources. Now, you’re given a unique RSA public key with a relatively small
key size (​64 bits​).

Your goal is to get the private key.

TODO:​ In the provided ​project_3.py​ file, implement the method ​get_factors​. is the givenn
public key, and your goal is to get its factors.

def​ ​get​_​factors​(self, n: int):
# TODO: Implement this method for Task 4, Step 1
p = ​0
q = ​0

​return​ p, q

TODO:​ In the provided ​project_3.py​ file, implement the method
get_private_key_from_p_q_e​ to get the private key.

def​ ​get_private_key_from_p_q_e​(self, p: int, q: int, e: int):
# TODO: Implement this method for Task 4, Step 2
d = ​0

​return​ d

GT​ CS 6035: Introduction to Information Security
Reflection
In a maximum of 500 words, address the following prompts:

Explain in your own words how you were able to get the private key. What were the steps you
followed in order to get it and what was the underlying mathematical principle used?
Task 5 – Where’s Waldo (​25​ Points)

which can be found at: ​https://factorable.net/weakkeys12.extended.pdf​. ​You will not be able to
entire paper​. Do not skip it, do not skim it, read the whole of it.

You are given a unique RSA public key, but the RNG (random number generator) used in the key
generation suffers from a vulnerability described in the paper above. In addition, you are given a list
of public keys that were generated by the same RNG on the same system. Your goal is to get the
unique private key from your given public key using only the provided information.

about Waldo, and why everyone keeps looking for him can be found here:
https://en.wikipedia.org/wiki/Where%27s_Wally%3F​. Knowledge of “Where’s Waldo?” isn’t strictly
necessary to solve this task, but it might give you a nudge in the right direction...)

given_public_key_n: int,
given_public_key_e: int,
public_key_list: list):
# TODO: Implement this method for Task 5
d = 0

​return​ d

Reflection
In a maximum of 500 words, address the following prompts:

the potential problems with the key generation and the associated mathematical principles in your

What steps did you take to derive the private key result in this task. Please discuss the underlying
mathematical principles at a high level and explain how you arrived at your answer.

A message was encrypted with three different 1,024-bit RSA public keys, resulting in three different
encrypted messages. All of them have the public exponent . 3e =

GT​ CS 6035: Introduction to Information Security

You are given the three pairs of public keys and associated encrypted messages. Your job is to
recover the original message.

TODO:​ In the provided ​project_3.py​ file, implement the method ​task_6​.

n_1_str: str, c_1_str: str,
n_2_str: str, c_2_str: str,
n_3_str: str, c_3_str: str):
n_1 = int(n_1_str, 16)
c_1 = int(c_1_str, 16)
n_2 = int(n_2_str, 16)
c_2 = int(c_2_str, 16)
n_3 = int(n_3_str, 16)
c_3 = int(c_3_str, 16)

msg = ​''
m = ​0

# Solve for m, which is an integer value,
# the line below will convert it to a string
msg = bytes.fromhex(hex(m).rstrip(​'​L​'​)[2:]).decode(​'​UTF-8​'​)

​return​ msg

Reflection
In a maximum of 500 words, address the following prompts:

How does the broadcast RSA attack work? What causes the vulnerability? Explain this in your own
words and explain at a high level the mathematical principles behind it.

Explain how you recovered the message, ensuring that you give thorough detail on all of your
steps.
Important Notes :

The skeleton code in the ​project_3.py​ file has all of the packages that you will need imported for
you. You are NOT allowed to import anything else.

Your entire submission must run in 10 minutes or less. ​The autograder will not give you any
feedback if it times out. We encourage you to test locally to avoid unnecessarily using
submissions.

GT​ CS 6035: Introduction to Information Security
submissions, penalties will be assessed as follows:

10 < # submissions <=20  -10
20 < # submissions <=30  -20
30 < # submissions <=40  -30
40 < # submissions <=50  -40
50 < # submissions <=60  -50
# submissions > 60  -55

You will be able to keep the score of your highest run.

do not wait until the last minute to make your submissions to the autograder. ​Try to get them in
as early as possible.

encourage you to read up on Python unit tests, but in general, the syntax should resemble either:

python -m unittest test_project_3

or:

python test_project_3.py

However, keep in mind that passing the unit test does NOT guarantee that your code will pass the
The final deliverables:

In total, please submit the following files:

1. ​project_3.py
2. ​project_3_report.pdf​ ​: An essay with all of your answers to the reflection questions.

Your written report must be submitted in the Joyner Document Format (JDF). A template has been
provided for you in Microsoft Word format, but you may find further useful resources here: