// Numbas version: finer_feedback_settings
{"name": "Vigen\u00e8re Cipher encryption/decryption", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Vigen\u00e8re Cipher encryption/decryption", "tags": [], "metadata": {"description": "<p>A straightforward test of encrypting and decrypting an eight-letter message with the Vigen&egrave;re Cipher using a four-letter key.&nbsp; Message letters are generated uniformly at random from the English alphabet, as are the key letters.&nbsp; Students are expected to be able to map English letters onto elements of $\\mathbb{Z}_{26}$ in the usual order.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Suppose we wish to encrypt the (nonsense) plaintext {message} using the Vigen&egrave;re Cipher.</p>", "advice": "<p>a) We convert each letter into an element of $\\mathbb{Z}_{26}$, so {message} becomes {numbermessage}.&nbsp; Similarly we convert the key {key} to {numberkey}.&nbsp; To encrypt the first four message numbers we add the corresponding key numbers, working modulo 26 so we obtain $[\\var{numbermessage[0]}+\\var{numberkey[0]},\\var{numbermessage[1]}+\\var{numberkey[1]},\\var{numbermessage[2]}+\\var{numberkey[2]},\\var{numbermessage[3]}+\\var{numberkey[3]}]$&nbsp; $\\equiv\\var{map(mod(numbermessage[i]+numberkey[i],26),i,0..3)}$.</p>\n<p>To encrypt the next four message numbers we then repeat that sequence of key numbers, which gives $[\\var{numbermessage[4]}+\\var{numberkey[0]},\\var{numbermessage[5]}+\\var{numberkey[1]},\\var{numbermessage[6]}+\\var{numberkey[2]}, \\var{numbermessage[7]}+\\var{numberkey[3]}]\\equiv\\var{map(mod(numbermessage[i]+numberkey[i-4],26),i,4..7)}$.</p>\n<p>b) Similarly we convert the ciphertext {ciphertext} into numerical form {numberciphertext}.&nbsp; To decrypt we now subtract the elements of {numberkey} from each element of the ciphertext in turn, again working modulo 26.&nbsp; We obtain $[\\var{numberciphertext[0]}-\\var{numberkey[0]},\\var{numberciphertext[1]}-\\var{numberkey[1]},\\var{numberciphertext[2]}-\\var{numberkey[2]},\\var{numberciphertext[3]}-\\var{numberkey[3]},\\var{numberciphertext[4]}-\\var{numberkey[0]},$$\\var{numberciphertext[5]}-\\var{numberkey[1]},\\var{numberciphertext[6]}-\\var{numberkey[2]},\\var{numberciphertext[7]}-\\var{numberkey[3]}]\\equiv\\var{map(mod(numberciphertext[i]-numberkey[mod(i,4)],26),i,0..7)}$.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"alphabet": {"name": "alphabet", "group": "Ungrouped variables", "definition": "\"ABCDEFGHIJKLMNOPQRSTUVWXYZ\"", "description": "", "templateType": "anything", "can_override": false}, "numbermessage": {"name": "numbermessage", "group": "Ungrouped variables", "definition": "Repeat(Random(0..25),8)", "description": "", "templateType": "anything", "can_override": false}, "message": {"name": "message", "group": "Ungrouped variables", "definition": "join(map(alphabet[x],x,numbermessage),\"\")", "description": "", "templateType": "anything", "can_override": false}, "numberkey": {"name": "numberkey", "group": "Ungrouped variables", "definition": "Repeat(Random(0..25),4)", "description": "", "templateType": "anything", "can_override": false}, "key": {"name": "key", "group": "Ungrouped variables", "definition": "join(map(alphabet[x],x,numberkey),\"\")", "description": "", "templateType": "anything", "can_override": false}, "ciphertext": {"name": "ciphertext", "group": "Ungrouped variables", "definition": "join(map(alphabet[x],x,numberciphertext),\"\")", "description": "", "templateType": "anything", "can_override": false}, "numberciphertext": {"name": "numberciphertext", "group": "Ungrouped variables", "definition": "Repeat(Random(0..25),8)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["alphabet", "numbermessage", "message", "numberkey", "key", "ciphertext", "numberciphertext"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Encrypt the plaintext using the key {key}.&nbsp; (Express your answer as a sequence of elements of $\\mathbb{Z}_{26}$.)</p>", "correctAnswer": "matrix(map(mod(numbermessage[i]+numberkey[mod(i,4)],26),i,0..7))", "correctAnswerFractions": false, "numRows": 1, "numColumns": "8", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Decrypt the ciphertext {ciphertext} using the key {key}.</p>", "correctAnswer": "matrix(map(mod(numberciphertext[i]-numberkey[mod(i,4)],26),i,0..7))", "correctAnswerFractions": false, "numRows": 1, "numColumns": "8", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Maura Paterson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7001/"}]}]}], "contributors": [{"name": "Maura Paterson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7001/"}]}