- Level: International Baccalaureate
- Subject: Maths
- Word count: 5189
The purpose of this investigation is to explore the various properties and concepts of matrix cryptography.
Extracts from this document...
Introduction
Danny Aburas
Year 12 Maths Studies
Investigation
Matrix Cryptography
Topic: Working with Linear Equations and Matrices
Subtopics: 3.3 Matrices
3.4 The Inverse of a Matrix
A completed investigation should include:
- an introduction that outlines the problem to be explored, including its significance, its features, and the context
- the method required to find a solution, in terms of the mathematical model or strategy to be used
- the appropriate application of the mathematical model or strategy, including
- the generation or collection of relevant data and/or information, with details of the process of collection
- mathematical calculations and results, and appropriate representations
- the analysis and interpretation of results
- reference to the limitations of the original problem
- a statement of the results and conclusions in the context of the original problem
- appendices and a bibliography, as appropriate.
Learning Requirements | Assessment Design Criteria | Capabilities |
1. understand fundamental mathematical concepts, demonstrate mathematical skills, and apply routine mathematical procedures 2. use mathematics as a tool to analyse data and other information elicited from the study of situations taken from social, scientific, economic, or historical contexts 3. think mathematically by posing questions/problems, making and testing conjectures, and looking for reasons that explain the results 4. make informed and critical use of electronic technology to provide numerical results and graphical representations 5. communicate mathematically and present mathematical information in a variety of ways 6. work both individually and cooperatively in planning, organising, and carrying out mathematical activities. | Mathematical Knowledge and Skills and Their Application The specific features are as follows: MKSA1 Knowledge of content and understanding of mathematical concepts and relationships. MKSA2 Use of mathematical algorithms and techniques (implemented electronically where appropriate) to find solutions to routine and complex questions. MKSA3 Application of knowledge and skills to answer questions in applied and theoretical contexts. Mathematical Modelling and Problem-solving The specific features are as follows: MMP1 Application of mathematical models. MMP2 Development of solutions to mathematical problems set in applied and theoretical contexts. MMP3 Interpretation of the mathematical results in the context of the problem. MMP4 Understanding of the reasonableness and possible limitations of the interpreted results, and recognition of assumptions made. MMP5 Development and testing of conjectures, with some attempt at proof. MMP6 Contribution to group work. Communication of Mathematical Information The specific features are as follows: CMI1 Communication of mathematical ideas and reasoning to develop logical arguments. CMI2 Use of appropriate mathematical notation, representations, and terminology. | Communication Learning |
Performance Standards for Stage 2 Mathematical Studies
Mathematical Knowledge and Skills and Their Application | Mathematical Modelling and Problem-solving | Communication of Mathematical Information | |
A | Comprehensive knowledge of content and understanding of concepts and relationships. Appropriate selection and use of mathematical algorithms and techniques (implemented electronically where appropriate) to find efficient solutions to complex questions. Highly effective and accurate application of knowledge and skills to answer questions set in applied and theoretical contexts. | Development and effective application of mathematical models. Complete, concise, and accurate solutions to mathematical problems set in applied and theoretical contexts. Concise interpretation of the mathematical results in the context of the problem. In-depth understanding of the reasonableness and possible limitations of the interpreted results, and recognition of assumptions made. Development and testing of valid conjectures, with proof. Constructive and productive contribution to group work. | Highly effective communication of mathematical ideas and reasoning to develop logical arguments. Proficient and accurate use of appropriate notation, representations, and terminology. |
B | Some depth of knowledge of content and understanding of concepts and relationships. Use of mathematical algorithms and techniques (implemented electronically where appropriate) to find some correct solutions to complex questions. Accurate application of knowledge and skills to answer questions set in applied and theoretical contexts. | Attempted development and appropriate application of mathematical models. Mostly accurate and complete solutions to mathematical problems set in applied and theoretical contexts. Complete interpretation of the mathematical results in the context of the problem. Some depth of understanding of the reasonableness and possible limitations of the interpreted results, and recognition of assumptions made. Development and testing of reasonable conjectures, with substantial attempt at proof. Productive contribution to group work. | Effective communication of mathematical ideas and reasoning to develop mostly logical arguments. Mostly accurate use of appropriate notation, representations, and terminology. |
C | Generally competent knowledge of content and understanding of concepts and relationships. Use of mathematical algorithms and techniques (implemented electronically where appropriate) to find mostly correct solutions to routine questions. Generally accurate application of knowledge and skills to answer questions set in applied and theoretical contexts. | Appropriate application of mathematical models. Some accurate and generally complete solutions to mathematical problems set in applied and theoretical contexts. Generally appropriate interpretation of the mathematical results in the context of the problem. Some understanding of the reasonableness and possible limitations of the interpreted results, and some recognition of assumptions made. Development and testing of reasonable conjectures, with some attempt at proof. Some productive contribution to group work. | Appropriate communication of mathematical ideas and reasoning to develop some logical arguments. Use of generally appropriate notation, representations, and terminology, with some inaccuracies. |
D | Basic knowledge of content and some understanding of concepts and relationships. Some use of mathematical algorithms and techniques (implemented electronically where appropriate) to find some correct solutions to routine questions. Sometimes accurate application of knowledge and skills to answer questions set in applied or theoretical contexts. | Application of a mathematical model, with partial effectiveness. Partly accurate and generally incomplete solutions to mathematical problems set in applied or theoretical contexts. Attempted interpretation of the mathematical results in the context of the problem. Some awareness of the reasonableness and possible limitations of the interpreted results. Attempted development or testing of a reasonable conjecture. Superficial contribution to group work. | Some appropriate communication of mathematical ideas and reasoning. Some attempt to use appropriate notation, representations, and terminology, with occasional accuracy. |
E | Limited knowledge of content. Attempted use of mathematical algorithms and techniques (implemented electronically where appropriate) to find limited correct solutions to routine questions. Attempted application of knowledge and skills to answer questions set in applied or theoretical contexts with limited effectiveness. | Attempted application of a basic mathematical model. Limited accuracy in solutions to one or more mathematical problems set in applied or theoretical contexts. Limited attempt at interpretation of the mathematical results in the context of the problem. Limited awareness of the reasonableness and possible limitations of the results. Limited attempt to develop or test a conjecture. Attempted contribution to group work. | Attempted communication of emerging mathematical ideas and reasoning. Limited attempt to use appropriate notation, representations, or terminology, and with limited accuracy. |
Middle
Results.
Encoding Method
A | B | C | D | E | F | G | H | I | J | K | L | M |
2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 |
N | O | P | Q | R | S | T | U | V | W | X | Y | Z |
1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 |
First, the following alphanumeric system was chosen. Note that the corresponding digits are not merely a direct substitution in order of letters, but instead, we can rearrange the corresponding digits in some pattern or random order. In this case, the first half of the alphabet is allocated even numbers, while the remaining half is allocated odd numbers. This renders the code tougher to break without the alphanumeric system being provided to the intended reader.
Now, a message, written in English, was created. The message chosen is “IONCANNONREADY”, a quote from the Command and Conquer game series, the ion cannon fires charged particles and causes devastating damage to the enemy base. This message translates to the corresponding numeric code using the system above :
18,3,1 ,6,2,1,1,3,1, 9,10,2,8,23.
This code must now be broken into uniform packets or “chunks” to be coded. It is decided that the chunks will consist of 4x1 matrices. This is as the encoding matrix is a 4x4 and to be multipliable by each packet, the number of columns of matrix A must equal the number of rows of matrix B by definition.
Notice that the two elements are blank*. To fix this, “dummy” letters will be placed into the packets at random only to complete all packets to 4x1 matrices.
Z, digit 25*, was positioned in places of empty elements at random. All packets are now 4x1 and the message matrices and are ready for encoding.
18, 3, 1, 6, 2, 1,1,3,1, 9,10,2,8,23 now becomes,18316211319102823**
18, 3, 1, 25, 6, 2, 1,1,3,1, 9, 10,2,8,23,25
183125*621131910282325*
Now the message is in a form that may be encoded, however, the message code at the present state is very straight forward as the digits directly represent the corresponding letters. A disguise of some kind must be used to give this code additional security by shifting all the values by some scalar quantity.
Conclusion
A is the original code.
X is the scramble matrix
B is the scrambled code matrix.
Using matrix algebra, it can be seen that the original code matrix, A, can be obtained by multiplying the scrambled matrix, B, by the inverse of the scramble matrix used, X-1.
AX=B
AXX-1=BX-1.
AI=BX-1.
A=BX-1
To simplify things a little, the inverse of the scramble matrix is provided instead of giving the scramble matrix and having the inverse found by the reader as this could help prevent errors.
The inverse of the scramble matrix is given. This is the key to decoding the code.
X-1= 1-619-59-531-9930813-82264-824-59373-12033759
- Multiply each of the four encrypted matrices above with X-1 (given above). graphic calculator should be used for this multiplication process. This process should result in with four packets of data that have been decoded.
- Even though the code has been decoded, the elements in the matrix code are cipher shifted and the process must be reversed. The following algorithm was used to cipher shift the elements of the matrices: 3x +6 = C,
where x is the original element value, and C is the cipher shifted value. The original value can be obtained by algebraically rearranging the cypher shift equation, using algebraic methods, to: C -63=x. Apply this algorithim to each element in the four matrices obtained in the previous step by inputing each element to the placeholder C and replace each value with X value obtained.
- Using the set of matrices obtained in the last step, starting with the first matrix, read the elements of each matrix systematically from top to bottom and list them in a line from right to left
- The alpha numeric code is provided bellow, to read this message, just replace each of the numbers with the corresponding letters of the alphabet. HAVE FUN!
A | B | C | D | E | F | G | H | I | J | K | L | M |
2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 |
N | O | P | Q | R | S | T | U | V | W | X | Y | Z |
1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 |
Decoding.
Using the decoding instructions, the code was decoded by a member of the class, Jordan Maguire.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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