Cracking the Code: Polynomial Encryption
Inquiry Framework
Question Framework
Driving Question
The overarching question that guides the entire project.How can we, as aspiring cryptographers, utilize the properties of polynomial functions and systems of equations to design unbreakable encryption methods and then rigorously test their vulnerabilities?Essential Questions
Supporting questions that break down major concepts.- How can polynomial functions be used to create encryption codes?
- How do long division and synthetic division help in cracking the code?
- What role do systems of linear equations and inequalities play in code-breaking?
- How can mathematical principles be applied to create and break encryption codes?
Standards & Learning Goals
Learning Goals
By the end of this project, students will be able to:- Understand and apply polynomial functions in creating encryption codes.
- Utilize long division and synthetic division to decrypt codes.
- Apply systems of linear equations and inequalities in code-breaking techniques.
- Evaluate the strengths and vulnerabilities of different encryption methods.
Entry Events
Events that will be used to introduce the project to studentsCybersecurity Breach Simulation
A mock cybersecurity firm reveals they've suffered a data breach, with key files encrypted using polynomial-based codes. Students, acting as cybersecurity analysts, must use their knowledge of polynomial functions and equation-solving to recover the data, highlighting the real-world applications of encryption in data protection.Portfolio Activities
Portfolio Activities
These activities progressively build towards your learning goals, with each submission contributing to the student's final portfolio.Polynomial Architect
Students will explore the basics of polynomial functions and their potential use in creating encryption codes.Steps
Here is some basic scaffolding to help students complete the activity.Final Product
What students will submit as the final product of the activityA detailed report outlining the characteristics of different polynomial functions and the rationale for choosing a specific function for encryption.Alignment
How this activity aligns with the learning objectives & standardsLearning Goal: Understand and apply polynomial functions in creating encryption codes.The Division Decoder
Students will learn how to use long division and synthetic division to decrypt codes created with polynomial functions.Steps
Here is some basic scaffolding to help students complete the activity.Final Product
What students will submit as the final product of the activityA step-by-step guide on how to use long division and synthetic division to decrypt codes, including examples and explanations.Alignment
How this activity aligns with the learning objectives & standardsLearning Goal: Utilize long division and synthetic division to decrypt codes.Linear Systems Cracker
Students will explore how systems of linear equations and inequalities can be used to break encryption codes.Steps
Here is some basic scaffolding to help students complete the activity.Final Product
What students will submit as the final product of the activityA presentation demonstrating how systems of linear equations and inequalities can be used to decrypt codes, with examples and explanations.Alignment
How this activity aligns with the learning objectives & standardsLearning Goal: Apply systems of linear equations and inequalities in code-breaking techniques.Encryption Architect & Vulnerability Tester
Students will design their own encryption methods using polynomial functions and test their vulnerabilities.Steps
Here is some basic scaffolding to help students complete the activity.Final Product
What students will submit as the final product of the activityA comprehensive report detailing the design, testing, and analysis of an encryption method, including recommendations for improving its security.Alignment
How this activity aligns with the learning objectives & standardsLearning Goal: Evaluate the strengths and vulnerabilities of different encryption methods.Rubric & Reflection
Portfolio Rubric
Grading criteria for assessing the overall project portfolioCracking the Code: Polynomial Encryption Portfolio Rubric
Polynomial Architect: Function Characteristics & Rationale
This category assesses the student's understanding of polynomial function characteristics and their ability to justify their choice of a specific function for encryption.Understanding of Polynomial Characteristics
Demonstrates understanding of polynomial function characteristics (degree, leading coefficient, end behavior) and their relevance to encryption.
Exemplary
4 PointsDemonstrates a sophisticated and nuanced understanding of polynomial characteristics, explaining their impact on encryption with clarity and precision. Provides insightful examples.
Proficient
3 PointsDemonstrates a thorough understanding of polynomial characteristics and their relevance to encryption. Explains concepts clearly with supporting examples.
Developing
2 PointsShows an emerging understanding of polynomial characteristics, but explanations may lack depth or clarity. Examples are limited or not fully relevant.
Beginning
1 PointsShows a limited or inaccurate understanding of polynomial characteristics. Struggles to explain their relevance to encryption. Provides minimal or irrelevant examples.
Rationale for Function Choice
Provides a clear and well-reasoned rationale for choosing a specific polynomial function for encryption, linking its characteristics to security features.
Exemplary
4 PointsProvides an exceptionally clear, insightful, and well-supported rationale for the chosen function, demonstrating a deep understanding of its strengths and weaknesses in an encryption context. Explains how these properties contribute to code security or vulnerability.
Proficient
3 PointsProvides a clear and well-reasoned rationale for the chosen function, linking its characteristics to potential security features. The explanation is logical and supported by evidence.
Developing
2 PointsProvides a rationale for the chosen function, but the reasoning may be incomplete or unclear. The connection between function characteristics and security features is not fully developed.
Beginning
1 PointsProvides a weak or illogical rationale for the chosen function. Fails to adequately connect function characteristics to security features. The explanation is minimal or missing.
The Division Decoder: Decryption Process & Effectiveness
This category assesses the student's ability to use long division and synthetic division to decrypt codes and analyze the effectiveness of each method.Application of Division Methods
Correctly applies long division and synthetic division to decrypt simple codes, showing accuracy and procedural fluency.
Exemplary
4 PointsDemonstrates mastery in applying both long division and synthetic division to accurately decrypt complex codes. Explains the steps clearly and efficiently, highlighting nuances and potential pitfalls.
Proficient
3 PointsCorrectly applies both long division and synthetic division to decrypt simple codes. Shows a good understanding of the procedures with minimal errors.
Developing
2 PointsApplies long division and synthetic division to decrypt codes, but with some errors or inconsistencies in the process. Understanding of the procedures is developing.
Beginning
1 PointsStruggles to apply long division and synthetic division to decrypt codes. Demonstrates a limited understanding of the procedures and makes frequent errors.
Analysis of Method Effectiveness
Analyzes the effectiveness of long division and synthetic division in code-breaking, comparing their strengths, weaknesses, and suitability for different types of codes.
Exemplary
4 PointsProvides a comprehensive and insightful analysis of the effectiveness of both long division and synthetic division in various code-breaking scenarios. Accurately identifies the strengths, weaknesses, and specific situations where each method excels. Justifies the analyses with concrete examples.
Proficient
3 PointsAnalyzes the effectiveness of long division and synthetic division in code-breaking, comparing their strengths and weaknesses. Provides a clear and logical comparison.
Developing
2 PointsAttempts to analyze the effectiveness of long division and synthetic division, but the comparison is superficial or lacks specific details. The discussion of strengths and weaknesses is limited.
Beginning
1 PointsFails to adequately analyze the effectiveness of long division and synthetic division. The comparison is minimal or missing, and there is little understanding of their strengths and weaknesses.
Linear Systems Cracker: Application & Limitations
This category assesses the student's ability to apply systems of linear equations and inequalities to decrypt codes and discuss the limitations of this approach.Application of Linear Systems
Effectively applies systems of linear equations and inequalities to decrypt codes, demonstrating accurate problem-solving skills.
Exemplary
4 PointsDemonstrates a sophisticated ability to apply systems of linear equations and inequalities to accurately decrypt complex codes. Explains the problem-solving process thoroughly, showing a deep understanding of the underlying mathematical principles and how they apply to code-breaking.
Proficient
3 PointsEffectively applies systems of linear equations and inequalities to decrypt codes. Shows a clear understanding of the methods and solves problems accurately.
Developing
2 PointsApplies systems of linear equations and inequalities to decrypt codes, but with some errors or inconsistencies in the process. Understanding of the methods is developing.
Beginning
1 PointsStruggles to apply systems of linear equations and inequalities to decrypt codes. Demonstrates a limited understanding of the methods and makes frequent errors.
Discussion of Limitations
Provides a thoughtful and insightful discussion of the limitations and advantages of using systems of equations in code-breaking, considering factors such as code complexity and computational requirements.
Exemplary
4 PointsPresents a comprehensive and nuanced discussion of the limitations and advantages of using systems of linear equations and inequalities in code-breaking. Considers a wide range of factors, including code complexity, computational requirements, and potential vulnerabilities, providing well-supported arguments and insightful conclusions.
Proficient
3 PointsProvides a thoughtful discussion of the limitations and advantages of using systems of equations in code-breaking. Considers relevant factors and provides a balanced perspective.
Developing
2 PointsAttempts to discuss the limitations and advantages of using systems of equations, but the discussion may be superficial or incomplete. Some relevant factors are mentioned, but not fully explored.
Beginning
1 PointsFails to adequately discuss the limitations and advantages of using systems of equations in code-breaking. The discussion is minimal or missing, and there is little understanding of the relevant factors.
Encryption Architect & Vulnerability Tester: Design, Testing & Analysis
This category assesses the student's ability to design an encryption method, test its vulnerabilities, and analyze its strengths and weaknesses.Encryption Method Design
Designs a creative and effective encryption method using polynomial functions, demonstrating a clear understanding of encryption principles.
Exemplary
4 PointsDesigns a highly creative, effective, and sophisticated encryption method using polynomial functions. Demonstrates an exceptionally clear and comprehensive understanding of encryption principles, incorporating advanced techniques to enhance security and resilience.
Proficient
3 PointsDesigns a creative and effective encryption method using polynomial functions. Demonstrates a clear understanding of encryption principles and implements a logical approach.
Developing
2 PointsDesigns an encryption method using polynomial functions, but the approach may be basic or lack creativity. Understanding of encryption principles is developing.
Beginning
1 PointsStruggles to design an encryption method using polynomial functions. Demonstrates a limited understanding of encryption principles and the approach is unclear or ineffective.
Vulnerability Testing & Analysis
Rigorously tests the encryption method for vulnerabilities, analyzes its strengths and weaknesses, and provides recommendations for improvement.
Exemplary
4 PointsConducts rigorous and exhaustive testing to identify vulnerabilities in the encryption method. Provides a detailed and insightful analysis of its strengths and weaknesses, including potential attack vectors. Offers innovative and practical recommendations for improving its security.
Proficient
3 PointsTests the encryption method for vulnerabilities, analyzes its strengths and weaknesses, and provides clear and reasonable recommendations for improvement. The analysis is well-supported by evidence.
Developing
2 PointsAttempts to test the encryption method for vulnerabilities, but the testing may be limited or superficial. The analysis of strengths and weaknesses is basic, and recommendations for improvement are limited.
Beginning
1 PointsFails to adequately test the encryption method for vulnerabilities. The analysis is minimal or missing, and there are few or no recommendations for improvement.