Problem 45279. Cryptography with A Square Matrix : Encoding

Created by Mehmet OZC

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{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","targetMode":"","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","targetMode":"","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Matrix inverse operation can be used to encode or decode of a message. For example</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ text = 'matlab & cody'.]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>numeric equivalent is</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ num = [109 97 116 108 97 98 32 38 32 99 111 100 121]]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>let</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ key = magic(3);]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>key</w:t></w:r><w:r><w:t> is an arbitrary square matrix and its inverse is exist.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>reshape the vector</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>num</w:t></w:r><w:r><w:t> so it has the same number of rows with</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>key</w:t></w:r><w:r><w:t> matrix</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ numMatrix = [109 108 32 99 121;\n 97 97 38 111 32;\n 116 98 32 100 32];]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Note that last two elements are filled with 32 which is the numeric equivalent of blank character ' '.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Finally multiply</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>key</w:t></w:r><w:r><w:t> with</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>numMatrix</w:t></w:r><w:r><w:t> (and reshape it) to obtain an encoded vector.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ encodedVector= [1665 1624 1541 1549 1495 1501 486 510 534 1503 1552 1595 1192 747 836]]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>if you multiply</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>inverse of key</w:t></w:r><w:r><w:t> with reshaped</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>encodedVector</w:t></w:r><w:r><w:t> you can obtain the ascii numbers of secret message.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Next Problem :</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/45282-cyrptography-with-a-square-matrix-decoding\"><w:r><w:t>Cryptography with A Square Matrix : Decoding</w:t></w:r></w:hyperlink></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Inspired from:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Barnett, R.A., Ziegler, M.R. and Byleen, K.E. (2015). Finite Mathematics for Business, Economics, Life Sciences, and Social Sciences. Pearson, 13th (Global) Edition. p:245-247.</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

Matrix inverse operation can be used to encode or decode of a message. For example<pre> text = 'matlab & cody'. </pre>numeric equivalent is<pre> num = [109 97 116 108 97 98 32 38 32 99 111 100 121]</pre>let<pre> key = magic(3);</pre>key is an arbitrary square matrix and its inverse is exist.reshape the vector num so it has the same number of rows with key matrix<pre> numMatrix = [109 108 32 99 121;
 97 97 38 111 32;
 116 98 32 100 32];</pre>Note that last two elements are filled with 32 which is the numeric equivalent of blank character ' '.Finally multiply key with numMatrix (and reshape it) to obtain an encoded vector.<pre> encodedVector= [1665 1624 1541 1549 1495 1501 486 510 534 1503 1552 1595 1192 747 836]</pre>if you multiply inverse of key with reshaped encodedVector you can obtain the ascii numbers of secret message.Next Problem : <a href = "https://www.mathworks.com/matlabcentral/cody/problems/45282-cyrptography-with-a-square-matrix-decoding">Cryptography with A Square Matrix : Decoding</a>Inspired from:Barnett, R.A., Ziegler, M.R. and Byleen, K.E. (2015). Finite Mathematics for Business, Economics, Life Sciences, and Social Sciences. Pearson, 13th (Global) Edition. p:245-247.

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Last Solution submitted on Feb 22, 2021

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Alfonso Nieto-Castanon on 19 Sep 2020

it's a bit unclear why padding is necessary in the first two test cases, where the number of characters (15) is already a multiple of the key size (3 or 5, respectively)

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Problem 45279. Cryptography with A Square Matrix : Encoding

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