diff --git a/assignments/a4/a4.tex b/assignments/a4/a4.tex
new file mode 100644
index 0000000..f9a01a7
--- /dev/null
+++ b/assignments/a4/a4.tex
@@ -0,0 +1,119 @@
+\documentclass[fontsize=11pt]{article}
+\usepackage{amsfonts}
+\usepackage{amsmath}
+\usepackage{amsthm}
+\usepackage[utf8]{inputenc}
+\usepackage[margin=0.75in]{geometry}
+
+\title{CSC110 Assignment 4: Number Theory, Cryptography, and Algorithm Running Time}
+\author{TODO: FILL IN YOUR NAME HERE}
+\date{\today}
+
+% Some useful LaTeX commands. You are free to use these or not, and also add your own.
+\newcommand{\N}{\mathbb{N}}
+\newcommand{\Z}{\mathbb{Z}}
+\newcommand{\R}{\mathbb{R}}
+\newcommand{\cO}{\mathcal{O}}
+\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
+
+\begin{document}
+\maketitle
+
+\section*{Part 1: Practice with Proofs}
+
+\begin{enumerate}
+
+\item[1.] Statement to prove:
+
+$$\forall a, k, n \in \Z,~ \gcd(a, n) = 1 \Rightarrow \gcd(a + kn, n) = 1$$
+
+\begin{proof}
+TODO: Your proof goes here.
+\end{proof}
+
+\item[2.] Statement to prove (we've expanded the definition of Omega for you!):
+
+$$\exists c, n_0 \in \R^+,~ \forall n \in \N,~ n \geq n_0 \Rightarrow \log_{3} n - \log_{11} n \geq c \cdot \log_{14} n$$
+
+\begin{proof}
+TODO: Your proof goes here.
+\end{proof}
+
+\item[3.] Statement to prove (we haven't expanded the definition of Big-O for you, but we encourage you to do so yourself):
+
+$$\forall f, g: \N \to \R^{\geq 0},~ g \in \cO(f) \land \big(\forall m \in \N,~ f(m) \geq 1 \big) \Rightarrow g \in \cO(\floor{f})$$
+
+\begin{proof}
+TODO: Your proof goes here.
+\end{proof}
+
+\end{enumerate}
+
+\newpage
+
+\section*{Part 2: Generating Coprime Numbers}
+
+\begin{enumerate}
+
+\item[1.]
+Not to be handed in.
+
+\item[2.]
+Complete this part in the provided \texttt{a4\_part2.py} starter file.
+Do \textbf{not} include your solution in this file.
+
+\item[3.]
+Prove that each loop invariant holds.
+
+\begin{enumerate}
+\item[a.] Loop Invariant 1
+\begin{proof}
+TODO: Your proof goes here.
+\end{proof}
+
+\item[b.] Loop Invariant 2
+\begin{proof}
+TODO: Your proof goes here.
+\end{proof}
+
+\item[c.] Loop Invariant 3
+\begin{proof}
+TODO: Your proof goes here.
+\end{proof}
+
+\item[d.] Loop Invariant 4
+\begin{proof}
+TODO: Your proof goes here.
+\end{proof}
+\end{enumerate}
+
+\item[4.]
+Complete this part in the provided \texttt{a4\_part2.py} starter file.
+Do \textbf{not} include your solution in this file.
+
+\item[5.]
+Complete this part in the provided \texttt{a4\_part2.py} starter file.
+Do \textbf{not} include your solution in this file.
+\end{enumerate}
+
+\newpage
+
+\section*{Part 3: Running-Time Analysis}
+
+\begin{enumerate}
+\item[1.]
+TODO: Running-time analysis of \texttt{coprime\_to\_2\_and\_3}.
+
+\item[2.]
+TODO: Running-time analysis of \texttt{starting\_coprime\_numbers}.
+
+\item[3.]
+TODO: Running-time analysis of \texttt{coprime\_to\_all}.
+\end{enumerate}
+
+\section*{Part 4: Two New Cryptosystems}
+
+Complete this part in the provided \texttt{a4\_part4.py} starter file.
+Do \textbf{not} include your solution in this file.
+
+\end{document}
diff --git a/assignments/a4/a4_part2.py b/assignments/a4/a4_part2.py
new file mode 100644
index 0000000..f057380
--- /dev/null
+++ b/assignments/a4/a4_part2.py
@@ -0,0 +1,118 @@
+"""CSC110 Fall 2021 Assignment 4, Part 2: Generating coprime numbers
+
+Instructions (READ THIS FIRST!)
+===============================
+Implement each of the functions in this file. As usual, do not change any function headers
+or preconditions. You do NOT need to add doctests.
+
+You may create additional helper functions to help break up your code into smaller parts.
+
+Copyright and Usage Information
+===============================
+
+This file is provided solely for the personal and private use of students
+taking CSC110 at the University of Toronto St. George campus. All forms of
+distribution of this code, whether as given or with any changes, are
+expressly prohibited. For more information on copyright for CSC110 materials,
+please consult our Course Syllabus.
+
+This file is Copyright (c) 2021 David Liu, Mario Badr, and Tom Fairgrieve.
+"""
+import math
+
+
+def coprime_to_2_and_3(n: int) -> list[int]:
+    """Return the natural numbers less than n that are coprime to both 2 and 3.
+
+    The returned list is sorted.
+
+    Preconditions:
+      - n >= 6
+
+    >>> coprime_to_2_and_3(20)
+    [1, 5, 7, 11, 13, 17, 19]
+
+    Implementation note: recall negative list indexing from Assignment 3.
+    For all lists lst and integers i between 0 and len(lst) - 1 inclusive,
+    lst[-i] == lst[len(lst) - i].
+    """
+    nums_so_far = [1, 5]
+    while nums_so_far[-2] + 6 < n:
+        # Note: Write four assert statements here expressing the four loop invariants from the
+        # assignment handout. These statements should be at the top of the loop body.
+
+        next_number = nums_so_far[-2] + 6
+        list.append(nums_so_far, next_number)
+
+    return nums_so_far
+
+
+def coprime_to_all(primes: set[int], n: int) -> list[int]:
+    """Return the positive integers less than n that are coprime to every number in primes.
+
+    The returned list is sorted.
+
+    Pay attention to the preconditions, as they are designed to help simplify your work for this
+    question.
+
+    Preconditions:
+        - primes != set()
+        - every element of primes is prime
+        - n >= math.prod(primes)
+
+    >>> coprime_to_all({2, 3}, 20)
+    [1, 5, 7, 11, 13, 17, 19]
+    >>> coprime_to_all({2, 3, 7}, 50)
+    [1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47]
+
+    Implementation notes:
+        - You MUST use the provided helper function starting_coprime_numbers in your implementation,
+          and may NOT modify it (even though it is not as efficient as it could be!!).
+        - You will find the math.prod function useful.
+    """
+
+
+def starting_coprime_numbers(primes: set[int]) -> list[int]:
+    """Return the numbers up to the product of the given primes that are coprime to all of them.
+
+    Note: the length of the returned list is is exactly equal to phi(math.prod(primes)), where
+    phi is the Euler totient function.
+
+    Preconditions:
+        - primes != set()
+        - every element of primes is prime
+
+    >>> starting_coprime_numbers({2, 3})
+    [1, 5]
+    >>> starting_coprime_numbers({3, 11})
+    [1, 2, 4, 5, 7, 8, 10, 13, 14, 16, 17, 19, 20, 23, 25, 26, 28, 29, 31, 32]
+    """
+    nums_so_far = []
+    m = math.prod(primes)
+
+    for k in range(1, m):
+        is_coprime = True
+        for p in primes:
+            if k % p == 0:
+                is_coprime = False
+        if is_coprime:
+            list.append(nums_so_far, k)
+
+    return nums_so_far
+
+
+if __name__ == '__main__':
+    # When you are ready to check your work with python_ta, uncomment the following lines.
+    # (Delete the "#" and space before each line.)
+    # IMPORTANT: keep this code indented inside the "if __name__ == '__main__'" block
+    # Leave this code uncommented when you submit your files.
+    # import python_ta
+    #
+    # python_ta.check_all(config={
+    #     'extra-imports': ['python_ta.contracts', 'math'],
+    #     'max-line-length': 100,
+    #     'disable': ['R1705', 'C0200']
+    # })
+
+    import doctest
+    doctest.testmod()
diff --git a/assignments/a4/a4_part4.py b/assignments/a4/a4_part4.py
new file mode 100644
index 0000000..133a2ab
--- /dev/null
+++ b/assignments/a4/a4_part4.py
@@ -0,0 +1,182 @@
+"""CSC110 Fall 2021 Assignment 4, Part 4: Two New Cryptosystems
+
+Instructions (READ THIS FIRST!)
+===============================
+Implement each of the functions in this file. As usual, do not change any function headers
+or preconditions. You do NOT need to add doctests.
+
+You may create some additional helper functions to help break up your code into smaller parts.
+
+Copyright and Usage Information
+===============================
+
+This file is provided solely for the personal and private use of students
+taking CSC110 at the University of Toronto St. George campus. All forms of
+distribution of this code, whether as given or with any changes, are
+expressly prohibited. For more information on copyright for CSC110 materials,
+please consult our Course Syllabus.
+
+This file is Copyright (c) 2021 David Liu, Mario Badr, and Tom Fairgrieve.
+"""
+
+
+################################################################################
+# Task 1 - The Grid Transpose Cryptosystem
+################################################################################
+def grid_encrypt(k: int, plaintext: str) -> str:
+    """Encrypt the given plaintext using the grid cryptosystem.
+
+    Preconditions:
+        - k >= 1
+        - len(plaintext) % k == 0
+        - plaintext != ''
+
+    >>> grid_encrypt(8, 'DAVID AND MARIO TEACH COMPUTER SCIENCE!!')
+    'DDTMCA EPIVMAUEIACTNDRHEC I REAOC !N OS!'
+    """
+
+
+def grid_decrypt(k: int, ciphertext: str) -> str:
+    """Decrypt the given ciphertext using the grid cryptosystem.
+
+    Preconditions:
+        - k >= 1
+        - len(ciphertext) % k == 0
+        - ciphertext != ''
+
+    >>> grid_decrypt(8, 'DDTMCA EPIVMAUEIACTNDRHEC I REAOC !N OS!')
+    'DAVID AND MARIO TEACH COMPUTER SCIENCE!!'
+    """
+
+
+def plaintext_to_grid(k: int, plaintext: str) -> list[list[str]]:
+    """Return the grid with k columns from the given plaintext.
+
+    Preconditions:
+        - k >= 1
+        - len(plaintext) % k == 0
+        - plaintext != ''
+    """
+
+
+def grid_to_ciphertext(grid: list[list[str]]) -> str:
+    """Return the ciphertext corresponding to the given grid.
+
+    Preconditions:
+        - grid != []
+        - grid[0] != []
+        - all({len(row1) == len(row2) for row1 in grid for row2 in grid})
+    """
+
+
+def ciphertext_to_grid(k: int, ciphertext: str) -> list[list[str]]:
+    """Return the grid corresponding to the given ciphertext.
+
+    Note that this grid should be the one that is used to generate the ciphertext.
+
+    Preconditions:
+        - k >= 1
+        - len(ciphertext) % k == 0
+        - ciphertext != ''
+    """
+
+
+def grid_to_plaintext(grid: list[list[str]]) -> str:
+    """Return the plaintext message corresponding to the given grid.
+
+
+    Preconditions:
+        - grid != []
+        - grid[0] != []
+        - all({len(row1) == len(row2) for row1 in grid for row2 in grid})
+    """
+
+
+################################################################################
+# Task 2 - Breaking The Grid Transpose Cryptosystem
+################################################################################
+def grid_break(ciphertext: str, candidates: set[str]) -> set[int]:
+    """Return the set of possible secret keys that decrypt the given ciphertext into a message
+    that contains at least one of the candidate words.
+
+    >>> candidate_words = {'DAVID', 'MINE'}
+    >>> grid_break('DDTMCA EPIVMAUEIACTNDRHEC I REAOC !N OS!', candidate_words) == {8, 10}
+    True
+    """
+
+
+def run_example_break(ciphertext_file: str, candidates: set[str]) -> list[str]:
+    """Return a list of possible plaintexts for the ciphertext found in the given file.
+
+    Based on the A4 directory structure, you can call this function like this:
+        >>> possible_plaintexts = run_example_break('ciphertexts/grid_ciphertext1.txt', {'climate'})
+    """
+    with open(ciphertext_file, encoding='utf-8') as f:
+        ciphertext = f.read()
+
+    # (Not to be handed in) Try completing this function by calling grid_break and returning a
+    # list of the possible plaintext messages.
+
+    return [ciphertext] + list(candidates)  # This is a dummy line, please replace it!
+
+
+################################################################################
+# Task 3 - The Permuted Grid Transpose Cryptosystem
+################################################################################
+def permutation_grid_encrypt(k: int, perm: list[int], plaintext: str) -> str:
+    """Encrypt the given plaintext using the grid cryptosystem.
+
+    Preconditions:
+        - k >= 1
+        - len(plaintext) % k == 0
+        - sorted(perm) == list(range(0, k))
+        - plaintext != ''
+
+    >>> permutation_grid_encrypt(8, [0, 1, 2, 3, 4, 5, 6, 7],
+    ...                          'DAVID AND MARIO TEACH COMPUTER SCIENCE!!')
+    'DDTMCA EPIVMAUEIACTNDRHEC I REAOC !N OS!'
+    >>> permutation_grid_encrypt(8, [3, 2, 5, 0, 7, 1, 6, 4],
+    ...                          'DAVID AND MARIO TEACH COMPUTER SCIENCE!!')
+    'IACTNVMAUE I REDDTMCN OS!A EPIAOC !DRHEC'
+    """
+
+
+def permutation_grid_decrypt(k: int, perm: list[int], ciphertext: str) -> str:
+    """Return the grid corresponding to the given ciphertext.
+
+    Note that this grid should be the one that is used to generate the ciphertext.
+
+    Preconditions:
+        - k >= 1
+        - len(ciphertext) % k == 0
+        - sorted(perm) == list(range(0, k))
+        - ciphertext != ''
+
+    >>> permutation_grid_decrypt(8, [0, 1, 2, 3, 4, 5, 6, 7],
+    ...                          'DDTMCA EPIVMAUEIACTNDRHEC I REAOC !N OS!')
+    'DAVID AND MARIO TEACH COMPUTER SCIENCE!!'
+    >>> permutation_grid_decrypt(8, [3, 2, 5, 0, 7, 1, 6, 4],
+    ...                          'IACTNVMAUE I REDDTMCN OS!A EPIAOC !DRHEC')
+    'DAVID AND MARIO TEACH COMPUTER SCIENCE!!'
+    """
+
+
+if __name__ == '__main__':
+    # When you are ready to check your work with python_ta, uncomment the following lines.
+    # (Delete the "#" and space before each line.)
+    # IMPORTANT: keep this code indented inside the "if __name__ == '__main__'" block
+    # Leave this code uncommented when you submit your files.
+    #
+    # import python_ta
+    # python_ta.check_all(config={
+    #     'extra-imports': ['python_ta.contracts'],
+    #     'allowed-io': ['run_example_break'],
+    #     'max-line-length': 100,
+    #     'disable': ['R1705', 'C0200']
+    # })
+
+    import python_ta.contracts
+    python_ta.contracts.check_all_contracts()
+
+    import doctest
+    doctest.testmod()
diff --git a/assignments/a4/ciphertexts/grid_ciphertext1.txt b/assignments/a4/ciphertexts/grid_ciphertext1.txt
new file mode 100644
index 0000000..769b354
--- /dev/null
+++ b/assignments/a4/ciphertexts/grid_ciphertext1.txt
@@ -0,0 +1 @@
+Ea hadc’rvlstai hnmo'carsetb  eica tlne idrtm ahar ate-tet   rachenhaadast n ,ogc fehw  aihtntuhghmee a dtna h mtecoh iuravnobiturl guiohpzfota u tsteio nolhdnai .rso  tfMeo onrtseyhtr.e g  oyJlf ua osstutthr  e iispncel e at cnhaleegit em l araatesbecto e uci6thv5 ae01ns,1g.0,e 07s(00 W 0aey rbeye ae sraaosrtu strt rchaieegb:rou e th methadtar pvktseio: n /bgv/e ecetrlnhyi e ms saebmtveaeegl.niln n acnvsyiaacnr.lgige aosotv fi/o oeftnv hsige dl ieamnnco cidEeaea/lrr) nt
diff --git a/assignments/a4/ciphertexts/grid_ciphertext2.txt b/assignments/a4/ciphertexts/grid_ciphertext2.txt
new file mode 100644
index 0000000..ca8347e
--- /dev/null
+++ b/assignments/a4/ciphertexts/grid_ciphertext2.txt
@@ -0,0 +1 @@
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diff --git a/assignments/a4/ciphertexts/permuted_ciphertext.txt b/assignments/a4/ciphertexts/permuted_ciphertext.txt
new file mode 100644
index 0000000..039d04b
--- /dev/null
+++ b/assignments/a4/ciphertexts/permuted_ciphertext.txt
@@ -0,0 +1 @@
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