Linear Codebreakers: Designing Secure Digital Encryption

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Linear Codebreakers: Designing Secure Digital Encryption

Grade 7Math10 days

Students take on the role of cryptographers to design and implement multi-layered digital encryption systems using the power of linear algebra. By mapping alphanumeric characters to numerical data, they apply linear functions (y = mx + b) to transform messages and disrupt frequency analysis patterns. Through algebraic manipulation, inverse operations, and the solving of systems of linear equations, students learn how to both secure and recover sensitive data in a simulated high-stakes environment.

CryptographyLinear FunctionsSystems Of EquationsInverse OperationsData TransformationFrequency AnalysisAlgebraic Manipulation

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Inquiry Framework

Question Framework

Driving Question

The overarching question that guides the entire project.How can we, as cryptographers, design a multi-layered encryption system using linear functions and systems of equations to protect our digital messages from frequency analysis?

Essential Questions

Supporting questions that break down major concepts.

How can we translate characters and symbols into mathematical variables to represent information as data?
In what ways do linear functions (y = mx + b) act as a 'key' to transform and hide original messages?
How does the slope and y-intercept of a linear transformation determine the difficulty of cracking an encryption?
How can we use systems of linear equations to create multi-layered codes that are resistant to frequency analysis?
Why is frequency analysis an effective tool for codebreakers, and how can mathematical transformations disrupt these patterns?
How do we manipulate and solve linear equations to reverse an encryption and accurately recover the original data?

Standards & Learning Goals

Learning Goals

By the end of this project, students will be able to:

Translate alphanumeric characters into numerical data sets to facilitate mathematical processing and encryption.
Construct and apply linear functions (y = mx + b) to transform original data into ciphertext, serving as the encryption key.
Analyze the effect of slope and y-intercept on data sets to determine how specific mathematical transformations obscure patterns from frequency analysis.
Design and solve systems of linear equations to create multi-layered encryption protocols that require simultaneous solutions for decryption.
Apply algebraic manipulation and inverse operations to rearrange encryption formulas, allowing for the accurate recovery of original messages.

Common Core State Standards for Mathematics

HSA-CED.A.2

Primary

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.Reason: Students will create linear equations that define the relationship between the original character (input) and the encrypted character (output).

HSA-REI.C.6

Primary

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.Reason: The project requires students to use systems of equations for multi-layered encryption, necessitating the ability to solve for variables to decrypt messages.

HSA-CED.A.4

Secondary

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.Reason: To decrypt a message, students must rearrange their linear encryption formula to solve for the original variable (the 'x' or input).

HSA-CED.A.3

Secondary

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.Reason: Students must work within the constraints of the character set (e.g., modulo arithmetic or a set range of integers) when designing their codes.

HSN-Q.A.2

Supporting

Define appropriate quantities for the purpose of descriptive modeling.Reason: The initial phase of the project requires defining how characters (quantities) are mapped to numbers for the encryption model.

HSF-IF.C.7.a

Supporting

Graph linear and quadratic functions and show intercepts, maxima, and minima.Reason: Graphing the linear transformations helps students visualize how the 'key' shifts or scales the data set to hide frequency patterns.

Entry Events

Events that will be used to introduce the project to students

The 'Glitch' in the Feed: The Failure of Simple Shifts

Students enter the room to find a projected 'live' social media feed that appears to have been intercepted by a hacker. After easily cracking a simple Caesar cipher (a basic shift), they encounter a second, 'unbreakable' message where the letter 'E' appears as different numbers every time, forcing them to realize a simple shift isn't enough—they need a mathematical rule (a linear function) to hide the patterns.

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Portfolio Activities

These activities progressively build towards your learning goals, with each submission contributing to the student's final portfolio.

Activity 1

The System Safeguard: Multi-Layered Security

To create an 'unbreakable' code, students will now implement a Multi-Layered Security Protocol. They will work in pairs to create a system of two linear equations. To decrypt a single letter, the recipient must find the intersection of two lines (the system's solution). This simulates modern high-level encryption where multiple 'keys' are needed to unlock data.

Steps

Here is some basic scaffolding to help students complete the activity.

1. Partner with another student. Each person provides one linear equation from Activity 2.

2. Create a 'Double-Locked' character where the 'x' and 'y' values must satisfy both equations simultaneously.

3. Solve the system of equations using the Substitution or Elimination method to find the point of intersection (x, y).

4. Verify the solution by graphing both lines on a single coordinate plane and identifying the intersection point.

5. Explain why a system of equations is harder for a 'hacker' to break than a single linear function.

Final Product

What students will submit as the final product of the activityA 'Multi-Key' Cipher Packet consisting of two equations, a graph showing their intersection, and the recovered secret coordinate.

Alignment

How this activity aligns with the learning objectives & standardsHSA-REI.C.6: Students solve systems of linear equations to decode a message that has been protected by two simultaneous mathematical rules.

Activity 2

The Alphanumeric Atlas: Mapping Data to Numbers

Before any encryption can happen, the 'data' (letters of the alphabet) must be converted into 'quantities' (numbers). In this activity, students will design their own Alphanumeric Atlas, choosing how to represent letters, spaces, and punctuation as numerical values. They must consider constraints: should 'A' be 0 or 1? How do we handle the end of the alphabet? This establishes the foundation for mathematical modeling.

Steps

Here is some basic scaffolding to help students complete the activity.

1. Assign a unique integer to every letter of the alphabet (A-Z). For example, A=1, B=2, or A=0, B=1. Decide how to handle spaces (e.g., Space=27).

2. Write down a 'Secret Motto' (a 5-7 word phrase) that will be the message you protect throughout the project.

3. Use your Alphanumeric Legend to translate your Secret Motto into a raw data set (a list of numbers).

4. Document any 'rules' or constraints for your mapping, such as what happens if a number exceeds your set range.

Final Product

What students will submit as the final product of the activityA customized Alphanumeric Legend and a 'Data Map' where a short phrase is converted into a string of raw numbers.

Alignment

How this activity aligns with the learning objectives & standardsHSN-Q.A.2: This activity focuses on defining appropriate quantities for descriptive modeling by establishing a numerical scale for non-numerical data (letters).

Activity 3

The Linear Lock: Crafting the Encryption Rule

Students will now design their 'Linear Lock'—a linear function in the form of y = mx + b. The input (x) is the original number from their Atlas, and the output (y) is the encrypted value. Students will experiment with different slopes (m) and y-intercepts (b) to see how they transform their data set. They will apply their function to their raw data to create their first 'Ciphertext.'

Steps

Here is some basic scaffolding to help students complete the activity.

1. Choose a slope (m) and a y-intercept (b) for your encryption function. Ensure your slope is an integer greater than 1.

2. Apply the formula y = mx + b to each number in your raw data set from Activity 1.

3. Organize your results in a T-chart showing 'Input (x)' vs. 'Encrypted Output (y)'.

4. Identify the 'Domain' (possible original letters) and 'Range' (possible encrypted values) of your function.

Final Product

What students will submit as the final product of the activityAn Encryption Worksheet featuring the chosen linear function and the resulting Ciphertext (encrypted numbers).

Alignment

How this activity aligns with the learning objectives & standardsHSA-CED.A.2: Students create linear equations in two variables to represent the relationship between the original letter (input x) and the encrypted code (output y).

Activity 4

The Inverse Engineer: Building the Decryption Key

Encryption is useless if you can't get the message back! Students must act as 'Inverse Engineers' to create a decryption key. They will use algebraic manipulation to rearrange their equation y = mx + b to solve for x. This allows the recipient of the message to work backward from the ciphertext to the original text.

Steps

Here is some basic scaffolding to help students complete the activity.

1. Start with your encryption equation: y = mx + b.

2. Use inverse operations to isolate x. Step 1: Subtract 'b' from both sides. Step 2: Divide both sides by 'm'.

3. Write your new 'Decryption Formula' in the form x = (y - b) / m.

4. Test your Decryption Formula by plugging in your encrypted y-values to see if you get your original x-values back.

5. Translate the recovered x-values back into letters using your Atlas from Activity 1.

Final Product

What students will submit as the final product of the activityA Decryption Manual that shows the step-by-step algebraic derivation of the inverse function and the successfully recovered motto.

Alignment

How this activity aligns with the learning objectives & standardsHSA-CED.A.4: Students rearrange the linear encryption formula (y = mx + b) to solve for the input (x), creating the decryption key.

Activity 5

The Frequency Ghost: Visualizing Transformations

How does a codebreaker see patterns? Usually through frequency analysis. In this activity, students will graph their original message data and their encrypted data on the same coordinate plane. They will analyze how the slope 'stretches' the data and how the intercept 'shifts' it, effectively hiding the original letter patterns from someone trying to guess the message based on letter frequency.

Steps

Here is some basic scaffolding to help students complete the activity.

1. Create a coordinate plane where the x-axis represents the position of the letter in your message and the y-axis represents the numerical value.

2. Plot the 'Raw Data' points (Activity 1) in one color.

3. Plot the 'Encrypted Data' points (Activity 2) in a different color.

4. Draw the line of the function y = mx + b through the encrypted points.

5. Write a brief 'Security Report' explaining how the slope changed the 'shape' of your message data compared to the original.

Final Product

What students will submit as the final product of the activityA 'Transformation Comparison' Graph and a written analysis of how the linear change obscures the original message's pattern.

Alignment

How this activity aligns with the learning objectives & standardsHSF-IF.C.7.a: Students graph their linear functions and analyze how the slope and intercept visually alter the distribution of the message data.

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Rubric & Reflection

Portfolio Rubric

Grading criteria for assessing the overall project portfolio

The Linear Codebreakers: Cryptographic Assessment Rubric

Category 1

Mathematical Modeling and Algebra

This category evaluates the student's proficiency in translating qualitative data into quantitative models and manipulating linear equations to transform and recover information.

Criterion 1

Modeling & Function Construction

Assessment of the student's ability to create a consistent alphanumeric legend and a valid linear encryption function (y = mx + b).

Exemplary

4 Points

The Alphanumeric Atlas is comprehensive and flawlessly executed. The encryption function (y = mx + b) is sophisticated, with a clear and accurate identification of the domain and range. The resulting ciphertext is perfectly calculated based on the chosen rule.

Proficient

3 Points

The Alphanumeric Atlas is complete and consistent. The encryption function is correctly structured as y = mx + b, and the domain and range are correctly identified. The ciphertext contains minimal to no calculation errors.

Developing

2 Points

The Alphanumeric Atlas is functional but may have minor inconsistencies. The encryption function follows the y = mx + b format, but there are errors in identifying domain/range or in the ciphertext calculations.

Beginning

1 Points

The Alphanumeric Atlas is incomplete or inconsistent. The encryption function is incorrectly formatted or applied, leading to significant errors in the ciphertext.

Criterion 2

Inverse Operations & Decryption Accuracy

Assessment of the student's ability to use inverse operations to rearrange the encryption formula and accurately recover the original data.

Exemplary

4 Points

The student demonstrates mastery by flawlessly rearranging y = mx + b to solve for x, providing a clear step-by-step algebraic derivation. The decryption process is error-free and successfully recovers the entire original motto.

Proficient

3 Points

The student correctly uses inverse operations to solve for x. The decryption formula is accurate, and the original motto is recovered with only minor clerical errors.

Developing

2 Points

The student attempts to rearrange the formula but makes a procedural error (e.g., incorrect order of operations). The decryption process is partially successful but contains significant errors.

Beginning

1 Points

The student struggles to apply inverse operations to isolate x. The decryption formula is incorrect, making the original message unrecoverable.

Category 2

Multi-Layered Systems and Constraints

This category focuses on the student's ability to work with multiple constraints and find simultaneous solutions, mimicking modern high-level encryption.

Criterion 1

Systems of Equations & Security Analysis

Assessment of the student's ability to solve a system of linear equations using multiple methods (algebraic and graphical) to decode a multi-layered message.

Exemplary

4 Points

The student solves the system of equations with absolute precision using both substitution/elimination and graphing. The verification of the intersection point is clear, and the student provides a sophisticated explanation of why systems increase security.

Proficient

3 Points

The student correctly solves the system of equations and identifies the intersection point. Both the algebraic solution and the graph are accurate. The explanation of security is clear and logical.

Developing

2 Points

The student finds a solution to the system, but there are errors in either the algebraic work or the graph. The explanation of security is basic or incomplete.

Beginning

1 Points

The student is unable to solve the system of equations or create a representative graph. There is little to no explanation of how systems function as a security layer.

Category 3

Data Visualization and Cryptographic Analysis

This category evaluates the student's ability to visualize mathematical changes and communicate how those changes impact the vulnerability of the encrypted data.

Criterion 1

Graphical Analysis & Pattern Disruption

Assessment of the student's ability to graph linear data and analyze how mathematical transformations (slope and intercept) obscure patterns.

Exemplary

4 Points

Graphs are perfectly scaled, labeled, and provide a clear visual of the transformation. The Security Report offers a profound analysis of how the slope 'stretches' and the intercept 'shifts' the data to defeat frequency analysis.

Proficient

3 Points

Graphs are accurate and appropriately labeled. The written analysis correctly identifies how the linear function changes the distribution of the data to hide the original message pattern.

Developing

2 Points

Graphs are mostly accurate but may lack proper labeling or scale. The analysis of how transformations hide patterns is surface-level or partially incorrect.

Beginning

1 Points

Graphs are inaccurate or missing. The student cannot explain how the mathematical transformation relates to the concept of frequency analysis or security.

Category 4

Communication and Metacognition

This category assesses the clarity, organization, and professional communication of the mathematical processes used throughout the project.

Criterion 1

Documentation & Mathematical Communication

Assessment of the student's ability to document their process, use mathematical vocabulary correctly, and reflect on their growth as a cryptographer.

Exemplary

4 Points

The portfolio is exceptionally organized, using precise mathematical terminology throughout. The student provides insightful reflections on the challenges of encryption and the power of linear modeling.

Proficient

3 Points

The portfolio is well-organized and uses appropriate mathematical vocabulary. The student clearly explains the steps taken and shows a solid understanding of the project's goals.

Developing

2 Points

The portfolio is somewhat organized, but the use of mathematical vocabulary is inconsistent. Explanations are present but lack detail or clarity.

Beginning

1 Points

The portfolio is disorganized and lacks clear documentation of the mathematical process. Little to no mathematical vocabulary is used.

Reflection Prompts

End-of-project reflection questions to get students to think about their learning

Question 1

How confident do you feel in your ability to use linear functions ($y = mx + b$) to transform a message into a secure code and then use inverse operations to decrypt it?

Scale

Required

Question 2

Frequency analysis relies on finding patterns in how often certain letters appear. Explain how changing the slope ($m$) and the $y$-intercept ($b$) of your function makes it harder for a 'hacker' to use frequency analysis to crack your code.

Text

Required

Question 3

In Activity 4, you acted as an 'Inverse Engineer.' Why is it mathematically necessary to rearrange the formula to solve for $x$ in order to decrypt a message? What would happen if you didn't use the correct inverse operations?

Text

Required

Question 4

Which phase of the Linear Codebreakers project did you find the most mathematically challenging to complete?

Multiple choice

Optional

Options

Designing the Alphanumeric Atlas (mapping letters to numbers)

Applying the linear transformation ($y = mx + b$) to every character

Rearranging the equation to create the Decryption Key (Inverse Functions)

Solving the system of equations for the Multi-Layered Security phase

Graphing the transformations to visualize the 'Frequency Ghost'

Question 5

Now that you've designed your own encryption, how has this project changed your perspective on the security of the digital messages you send every day on social media or texting apps?

Text

Required

Question 6

Compare a single linear function to a system of two linear equations. In your opinion, why does the 'System Safeguard' (using two equations at once) provide significantly better security than a simple 'Linear Lock'?

Text

Optional