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"cc49d477-d61d-4251-825b-fb9e4bab2f75"], ")=x, which contradicts (*)!!!", StyleBox["\n", FontWeight->"Bold"], "\n\n\n", StyleBox["Exercise", FontWeight->"Bold"], ": Show that if |X|=|Y| (#X=#Y), then both ", Cell[BoxData[ FormBox[ SubscriptBox["e", "k"], TraditionalForm]],ExpressionUUID-> "0e5478c5-05ba-408d-963c-7b4e5783b62d"], " and ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["d", "k"], " "}], TraditionalForm]],ExpressionUUID-> "f8d33629-e7e8-4383-a3dc-69c0d909663a"], " are bijection.\n\ntry agian\n" }], "Text", CellChangeTimes->{{3.728063360166786*^9, 3.7280634936229763`*^9}, { 3.728063535362846*^9, 3.728063745563922*^9}, {3.759484188391281*^9, 3.759484189828113*^9}, {3.75949484384448*^9, 3.75949488119934*^9}, { 3.759494980196957*^9, 3.759495121586035*^9}, {3.759495168804051*^9, 3.759495344640409*^9}, {3.759495383559705*^9, 3.759495386549693*^9}, 3.790928614590165*^9, 3.790943941847095*^9, {3.79094443184658*^9, 3.7909444593508873`*^9}, {3.790944492792385*^9, 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From this point \ on we will use always + and \[Cross] in these rings." }], "Text", CellChangeTimes->{{3.728063786158483*^9, 3.728063939568638*^9}, { 3.72806475187954*^9, 3.7280647766300983`*^9}, {3.7280652206566067`*^9, 3.728065337300926*^9}, {3.728065404736391*^9, 3.728065785427725*^9}, { 3.728065835734226*^9, 3.728065858850994*^9}, {3.728065947974036*^9, 3.728065963512053*^9}, {3.7280660020474653`*^9, 3.72806603852906*^9}, { 3.728066081811809*^9, 3.728066086081491*^9}, {3.728066627661118*^9, 3.72806666622799*^9}},ExpressionUUID->"0a84d56a-e1f2-40f6-b080-\ 510d9998145e"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Mod", "[", RowBox[{ RowBox[{"-", "3"}], ",", "7"}], "]"}]], "Input", CellChangeTimes->{{3.728066039826393*^9, 3.728066049012512*^9}},ExpressionUUID->"a969e41e-084c-4403-ad20-\ 6921bc950279"], Cell[BoxData["4"], "Output", CellChangeTimes->{3.728066049549563*^9, 3.759495468184379*^9, 3.790944895276823*^9},ExpressionUUID->"558b74a4-7be9-46c5-ac94-\ 88b0a8a8f7e9"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"PowerMod", "[", RowBox[{"3", ",", "983475603498760349851", ",", "84576948579"}], "]"}]], "Input", CellChangeTimes->{{3.7594957787455063`*^9, 3.759495806091977*^9}, { 3.75949585736876*^9, 3.759495857598384*^9}, {3.79094494968069*^9, 3.790944957824759*^9}},ExpressionUUID->"019c6e1a-4800-44d9-8e88-\ 27fa88dd032f"], Cell[BoxData["51655902228"], "Output", CellChangeTimes->{{3.75949578782463*^9, 3.759495806572774*^9}, 3.759495858211623*^9, {3.790944953645226*^9, 3.7909449589657917`*^9}},ExpressionUUID->"7ceb54c0-578b-43a1-8c07-\ 02f64c66872d"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Mod", "[", RowBox[{ RowBox[{"5", "*", "3"}], ",", "7"}], "]"}]], "Input", CellChangeTimes->{{3.7280660521271353`*^9, 3.728066076961423*^9}},ExpressionUUID->"481bc3e7-2422-48b5-b6e8-\ c0b9399a28c1"], Cell[BoxData["1"], "Output", CellChangeTimes->{3.728066077358637*^9, 3.759495775506962*^9},ExpressionUUID->"9f8ccce0-1f1b-45e7-9cdb-\ e5bd08d457ac"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Mod", "[", RowBox[{"1239238470239", ",", "27"}], "]"}]], "Input", CellChangeTimes->{{3.75949619767486*^9, 3.759496203023192*^9}},ExpressionUUID->"5540b052-df21-4dee-a028-\ 6c43b08cbbc1"], Cell[BoxData["26"], "Output", CellChangeTimes->{ 3.759496208365223*^9},ExpressionUUID->"01c1f366-45ec-4f86-bc8a-\ ab6fe7379275"] }, Open ]], Cell[TextData[{ "\n", StyleBox["Exercise: ", FontWeight->"Bold"], " If 0"Bold"], "Note that", StyleBox["\n", FontWeight->"Bold"], "n=q*a+r with r"Bold"], "\n\n", StyleBox["Lemma: ", FontWeight->"Bold"], "Let d=gcd(a,n) then d| (k a +l n) ", Cell[BoxData[ FormBox[ SubscriptBox["\[ForAll]", RowBox[{"k", ",", RowBox[{"l", "\[Element]", "\[DoubleStruckCapitalZ]"}]}]], TraditionalForm]],ExpressionUUID->"c454fde0-f0c3-4e51-970d-4971c2ee2093"], ". Moreover, there exists an k and l such that d=(k a +l n).\n\nProof. By \ induction on (a,n)\n\n Not yet, because its the 1st day, and everybody looks \ scared, but it amounts to the soundness of the EEuclides algorithm\n\nWe are \ going be interested in the elements x in ", Cell[BoxData[ FormBox[ SubscriptBox["Z", "n"], TraditionalForm]],ExpressionUUID-> "b7115363-9794-4331-8c87-c684892545c4"], " such that x has an inverse.\n\nMeaning, there is a number y, such that x\ \[Cross]y = 1 mod n (y is the inverse of x mod n)\n\nx\[Cross]y +k n= 1 \n\n\ This can only hold if gcd(x,n)=1!!!\n\n\n\n\n", StyleBox["Note: ", FontWeight->"Bold"], "Mathematica has inbuilt the EE with the command ExtendedGCD\n" }], "Text", CellChangeTimes->{{3.728063786158483*^9, 3.728063939568638*^9}, { 3.72806475187954*^9, 3.7280647766300983`*^9}, {3.7280652206566067`*^9, 3.728065337300926*^9}, {3.728065404736391*^9, 3.728065785427725*^9}, { 3.728065835734226*^9, 3.728065858850994*^9}, {3.728065947974036*^9, 3.728065963512053*^9}, {3.7280660020474653`*^9, 3.7280660285595207`*^9}, 3.7280661029774437`*^9, {3.728066133415245*^9, 3.72806617667214*^9}, { 3.7280662214601183`*^9, 3.7280662452846127`*^9}, 3.728066693655623*^9, { 3.728128766405775*^9, 3.7281287687561274`*^9}, {3.7594841942202597`*^9, 3.759484200716187*^9}, {3.759495959134282*^9, 3.759495960723509*^9}, { 3.75949615416901*^9, 3.759496185879519*^9}, {3.75949623190954*^9, 3.759496307144742*^9}, {3.759496342722426*^9, 3.759496343447701*^9}, { 3.759496391950181*^9, 3.759496601837248*^9}, {3.75949666375595*^9, 3.7594966647322407`*^9}, {3.7594967476022882`*^9, 3.759496784266345*^9}, { 3.759496843304092*^9, 3.759496975446765*^9}, {3.7594970558561*^9, 3.759497056037973*^9}, {3.7909286409281807`*^9, 3.79092865006444*^9}, { 3.790943948475906*^9, 3.790943958892744*^9}, 3.790945302097921*^9, { 3.790945387594762*^9, 3.790945474463661*^9}, {3.790945504950603*^9, 3.790945560868194*^9}, {3.790945593945778*^9, 3.7909456872578373`*^9}, { 3.790945938539733*^9, 3.790945955960702*^9}, 3.823600626359817*^9, { 3.823600660931182*^9, 3.8236006830085793`*^9}, {3.8244663645464153`*^9, 3.824466368519375*^9}},ExpressionUUID->"9dae6203-6928-4b2a-8636-\ 806567c5734f"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ExtendedGCD", "[", RowBox[{"9384759048755", ",", "3049583094853049859"}], "]"}]], "Input", CellChangeTimes->{{3.728066177544704*^9, 3.728066187852833*^9}, 3.759497089930808*^9, {3.790945997939227*^9, 3.790946000500527*^9}},ExpressionUUID->"27a38579-776f-4f31-b416-\ 3f5d06147bf6"], Cell[BoxData[ RowBox[{"{", RowBox[{"1", ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1430813131668802090"}], ",", "4403171209589"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.728066188400116*^9, 3.75949709035255*^9, {3.790945980961875*^9, 3.790946002204813*^9}},ExpressionUUID->"e0275785-1a3b-4aa6-a8d8-\ cece6cc0c0cf"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ExtendedGCD", "[", RowBox[{"2342323423442334", ",", "7223234234242343410"}], "]"}]], "Input", CellChangeTimes->{{3.7280661953475037`*^9, 3.7280661966964293`*^9}, { 3.728066261129562*^9, 3.728066263876286*^9}, {3.7594971136677723`*^9, 3.7594971204887867`*^9}, {3.759497155801379*^9, 3.759497156372167*^9}, { 3.790946052711784*^9, 3.79094605290441*^9}},ExpressionUUID->"e0704379-4804-4038-90e2-\ 0e81dac56fd6"], Cell[BoxData[ RowBox[{"{", RowBox[{"2", ",", RowBox[{"{", RowBox[{"1292683643382161498", ",", RowBox[{"-", "419186624551213"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.728066264447709*^9, {3.75949711581887*^9, 3.759497121025386*^9}, 3.7594971571513853`*^9, 3.790946055824039*^9},ExpressionUUID->"ff20962e-ccff-4cf5-89c6-\ 19850525b796"] }, Open ]], Cell[TextData[{ StyleBox["Proposition 4", FontWeight->"Bold"], ". Let a \[Element] ", Cell[BoxData[ FormBox[ SubscriptBox["\[DoubleStruckCapitalZ]", "n"], TraditionalForm]], ExpressionUUID->"25e79414-60a9-4cba-b7d6-2cb1540c35b7"], " then a is invertible iff gcd(a,n)=1.\n\nProof: then there exists \ b\[Cross]a=1 mod n implies\n\nb\[Cross]a=k\[Cross]n +1 which means that b\ \[Cross]a+(-k)\[Cross]n=1 -> 1 is the gcd(a,n)\n\n1=gcd(a,n)=k\[Cross]a+l\ \[Cross]n \n1-l\[Cross]n=k\[Cross]a\n\nthis means that k\[Cross]a=1 mod n so \ k is the inverse of a mod\n\nThe proof follows from the extend Euclides \ algorithm\n\nSo, if a is invertible then there exists b such that ab=1 mod n, \ meaning that\n\nab+kn=1, and the smallest value possible to write as an \ integer combination of a and n is precisely gcd(a,n), thus gcd(a,n)=1\n\nif \ gcd(a,n)=1, by EEalgorithm , there exists k and l such that ak+ln=1 and so k \ is the inverse of a mod n" }], "Text", CellChangeTimes->{{3.7280658129465437`*^9, 3.728065830730647*^9}, { 3.7280658662637367`*^9, 3.728065875444409*^9}, {3.728065924474074*^9, 3.728065974102066*^9}, 3.728066100599947*^9, {3.728066303151218*^9, 3.728066482702569*^9}, {3.728066516276279*^9, 3.728066595185265*^9}, { 3.728066674829728*^9, 3.7280666792191477`*^9}, {3.728066714150642*^9, 3.72806674382896*^9}, {3.728066784963064*^9, 3.72806678555409*^9}, { 3.759484208354827*^9, 3.759484209681292*^9}, {3.759497177315907*^9, 3.759497277829335*^9}, {3.759497330552815*^9, 3.75949733610898*^9}, 3.7909439633715763`*^9, {3.790946116855364*^9, 3.790946142443952*^9}, { 3.7909462022473917`*^9, 3.79094625587372*^9}, {3.790946326620315*^9, 3.790946347739974*^9}, {3.790946387863688*^9, 3.7909464676417*^9}, { 3.823601303632901*^9, 3.823601303897444*^9}, {3.824466332998928*^9, 3.824466334113435*^9}},ExpressionUUID->"5d11553e-b910-4cf2-bd09-\ 75b51f6a8d80"], Cell["", "Text", CellChangeTimes->{{3.7280658129465437`*^9, 3.728065830730647*^9}, { 3.7280658662637367`*^9, 3.728065875444409*^9}, {3.728065924474074*^9, 3.728065974102066*^9}, {3.728066746170786*^9, 3.728066782243939*^9}},ExpressionUUID->"53873895-df05-44f7-81d0-\ c393cdf9b381"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ExtendedGCD", "[", RowBox[{"5", ",", "7"}], "]"}]], "Input", CellChangeTimes->{{3.728066779349444*^9, 3.728066788284831*^9}},ExpressionUUID->"3312b64a-ae9c-4ab1-acc5-\ 9719f5d79e57"], Cell[BoxData[ RowBox[{"{", RowBox[{"1", ",", RowBox[{"{", RowBox[{"3", ",", RowBox[{"-", "2"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{{3.728066780118009*^9, 3.728066789217483*^9}},ExpressionUUID->"9b2d9cc4-f424-44be-b720-\ 0a08d40d42f0"] }, Open ]], Cell[TextData[{ "Which is the inverse of 5 in ", Cell[BoxData[ FormBox[ SubscriptBox["\[DoubleStruckCapitalZ]", "7"], TraditionalForm]], ExpressionUUID->"fc36da3e-08bd-406b-b8d3-ce6e1c5d4dfe"], "?" }], "Text", CellChangeTimes->{{3.728066790047483*^9, 3.728066827040957*^9}}, FontWeight->"Plain",ExpressionUUID->"a4a44984-4a62-4192-8042-f8dad12ed68d"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"PowerMod", "[", RowBox[{"5", ",", RowBox[{"-", "1"}], ",", "7"}], "]"}]], "Input", CellChangeTimes->{{3.728066878889433*^9, 3.728066892171274*^9}},ExpressionUUID->"6214075c-837b-4998-9cb2-\ e26d631312e9"], Cell[BoxData["3"], "Output", CellChangeTimes->{ 3.728066892444487*^9},ExpressionUUID->"d546e552-a808-4ffd-9086-\ fdd5750e3cc1"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Affine and substitution cryptosystems", "Subsubsection", CellChangeTimes->{{3.728194881327486*^9, 3.72819489643839*^9}},ExpressionUUID->"4405dd61-e610-4d86-a35a-\ 26935e3792b7"], Cell[TextData[{ StyleBox["Definition 2. 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It is an ", StyleBox["open problem", FontWeight->"Bold"], " if this function can be computed in polynomial-time. If this is the case \ RSA can be broken!" }], "Text", CellChangeTimes->{{3.728194106959568*^9, 3.7281941680488*^9}, { 3.728194232014592*^9, 3.7281942523019447`*^9}, {3.823850722452859*^9, 3.8238507241345453`*^9}},ExpressionUUID->"df326afc-6802-4ddf-9cbd-\ b9bb27f7a773"], Cell[TextData[{ StyleBox["Exercise: ", FontWeight->"Bold"], "Show that (", Cell[BoxData[ FormBox[ SubsuperscriptBox["\[DoubleStruckCapitalZ]", "n", "\[Cross]"], TraditionalForm]],ExpressionUUID->"84412a9a-d7a4-4161-9e0e-7388fc6698a9"], ",\[Cross]) is indeed a group:\n(i) is closed under multiplication\n(ii) has \ an identity element.\n(iii) is closed under inverses\n\n(i) Assume ", StyleBox["a ", FontSlant->"Italic"], " and ", StyleBox["b \[Element]", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{" ", SubsuperscriptBox["\[DoubleStruckCapitalZ]", "n", "\[Cross]"]}], TraditionalForm]],ExpressionUUID->"04b97c66-5a80-4483-a2d5-9df70a065fbb"], ", we have to show that ", StyleBox["a", FontSlant->"Italic"], "\[Cross]", StyleBox["b\[Element]", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{" ", SubsuperscriptBox["\[DoubleStruckCapitalZ]", "n", "\[Cross]"]}], TraditionalForm]],ExpressionUUID->"5ffcc846-19a6-4aec-ad1b-c4f77a01c393"], StyleBox[", ", FontSlant->"Italic"], "that is, if ", StyleBox["a ", FontSlant->"Italic"], " and ", StyleBox["b ", FontSlant->"Italic"], "are invertible si is their multiplication. Clearly ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"a", "\[Cross]", "b"}], ")"}], RowBox[{"-", "1"}]], TraditionalForm]],ExpressionUUID-> "816e5435-9088-4806-ac8b-112e6a3bf9b7"], "= ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["b", RowBox[{"-", "1"}]], "\[Cross]", SuperscriptBox["a", RowBox[{"-", "1"}]]}], TraditionalForm]],ExpressionUUID-> "0b7640a2-dff0-4c24-b0c4-656b71844341"], " mod n, since (", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["b", RowBox[{"-", "1"}]], "\[Cross]", SuperscriptBox["a", RowBox[{"-", "1"}]]}], TraditionalForm]],ExpressionUUID-> "efb5098f-eeeb-46c1-8eb9-315841db3af3"], ")\[Cross](", StyleBox["a", FontSlant->"Italic"], "\[Cross]", StyleBox["b)=1 ", FontSlant->"Italic"], "(since modular multiplication is commutative, this is enough)\n\n(ii) so is \ 1 invertible, indeed the ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["1", RowBox[{"-", "1"}]], "=", "1"}], ",", " ", RowBox[{ RowBox[{"since", " ", RowBox[{"1", "\[Cross]", "1"}]}], "=", "1"}]}], TraditionalForm]], ExpressionUUID->"70bc8e57-0528-4174-955c-61778d36c4d1"], "\n\n(iii) if ", Cell[BoxData[ FormBox["a", TraditionalForm]],ExpressionUUID-> "f3c71bb2-f078-40a8-b575-1b3fbe2065d1"], " is invertible then ", Cell[BoxData[ FormBox[ SuperscriptBox["a", RowBox[{"-", "1"}]], TraditionalForm]],ExpressionUUID-> "42fa9d3e-4fd8-4da0-82d8-ecfb3cb6058b"], " is also invertible, as the inverse of ", Cell[BoxData[ FormBox[ SuperscriptBox["a", RowBox[{"-", "1"}]], TraditionalForm]],ExpressionUUID-> "3c57decc-31a6-4eec-af8a-b2cb9ecf781c"], " is ", StyleBox["a", FontSlant->"Italic"], " itself." }], "Text", CellChangeTimes->{{3.728194264667706*^9, 3.7281943079076223`*^9}, { 3.7282954808869333`*^9, 3.728295672787565*^9}, {3.7282957086488333`*^9, 3.728295732056284*^9}, {3.7282957693551054`*^9, 3.7282958655007486`*^9}, 3.791190968526723*^9, {3.791193399545845*^9, 3.791193400499915*^9}, { 3.791193460639101*^9, 3.791193652821847*^9}, {3.823850743198127*^9, 3.82385085024058*^9}, {3.8238508966801767`*^9, 3.823850897197942*^9}, { 3.8238530430590754`*^9, 3.823853438131331*^9}},ExpressionUUID->"4c01abaa-6c64-448b-82c7-\ 7d60d77372ed"], Cell[TextData[{ StyleBox["Exercise: ", FontWeight->"Bold"], "If n is prime what is the value of \[Phi](n)?\n\nIf n is prime the \ \[Phi](n)=n-1" }], "Text", CellChangeTimes->{{3.728194365004538*^9, 3.728194426223599*^9}, { 3.728194525405327*^9, 3.7281945365395193`*^9}, {3.7281945796910353`*^9, 3.7281945929938*^9}, {3.728194640082069*^9, 3.728194868092355*^9}, { 3.791193666722192*^9, 3.7911936669372168`*^9}, {3.7911937009914837`*^9, 3.791193708589446*^9}, {3.8238509118474617`*^9, 3.8238509123341084`*^9}, { 3.823853528699589*^9, 3.8238535364343863`*^9}},ExpressionUUID->"cd7cb182-497b-43f3-bc1c-\ 69d4d44e69bc"], Cell[TextData[{ StyleBox["\nDefinition 3", FontWeight->"Bold"], ". Affine Cryptosystem \[ScriptCapitalA]=", StyleBox["(X,Y,K,e,d)\[LineSeparator]", "DisplayFormula"], "\n\[Bullet] X=Y=", Cell[BoxData[ FormBox[ SubscriptBox["\[DoubleStruckCapitalZ]", "n"], TraditionalForm]], ExpressionUUID->"d5a851a0-4d6d-40d6-8088-a700c95fa6bf"], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Bullet]", " ", "K"}], " ", "=", RowBox[{ SubsuperscriptBox["\[DoubleStruckCapitalZ]", "n", "\[Cross]"], "\[Cross]", SubscriptBox["\[DoubleStruckCapitalZ]", "n"]}]}], TraditionalForm]], ExpressionUUID->"615dae18-a8ce-4655-b3e2-bf687498be41"], "\n\[Bullet] ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["e", RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}]], "(", "x", ")"}], "=", RowBox[{ RowBox[{"a", " ", "x"}], "+", " ", RowBox[{"b", " ", "mod", " ", "n"}]}]}], TraditionalForm]], ExpressionUUID->"ca0e93ab-2395-4878-8c1e-5c3b5ccd1439"], "\n\n", StyleBox["Exercise: ", FontWeight->"Bold"], "Show that if a is not invertible mod n this is not a cryptosystem (since ", Cell[BoxData[ FormBox[ SubscriptBox["e", RowBox[{"(", RowBox[{"a", ".", "b"}], ")"}]], TraditionalForm]],ExpressionUUID-> "8f06150f-c81e-41be-a6a0-ea8456208749"], " will not be injective)\n\n", StyleBox["Exercise", FontWeight->"Bold"], ": Show that ", Cell[BoxData[ FormBox[ SubscriptBox["e", RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}]], TraditionalForm]],ExpressionUUID-> "4b2ee94b-d1df-4d5e-b2eb-25be0a697d82"], "is a bijection (also called a ", StyleBox["permutation", FontSlant->"Italic"], ") and compute its inverse, that is, provide ", Cell[BoxData[ FormBox[ SubscriptBox["d", RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}]], TraditionalForm]],ExpressionUUID-> "93326a28-db0c-4056-aa99-72c11ad03736"], ".\n\nSince X=Y it is enough to show that it is injective.\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["e", RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}]], "(", "x", ")"}], "=", RowBox[{ SubscriptBox["e", RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}]], "(", RowBox[{"x", "'"}], ")"}]}], TraditionalForm]],ExpressionUUID-> "a46c985a-c398-47d0-8d70-546ebf6d38cf"], " \[RightArrow] ", StyleBox["a x+b=ax\[CloseCurlyQuote]+b mod n \[RightArrow] a x = a x\ \[CloseCurlyQuote] mod n \[RightArrow] ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["a", RowBox[{"-", "1"}]], "(", RowBox[{"a", " ", "x"}], ")"}], "=", RowBox[{ RowBox[{ SuperscriptBox["a", RowBox[{"-", "1"}]], "(", RowBox[{"a", " ", RowBox[{"x", "'"}]}], ")"}], " ", "mod", " ", "n"}]}], TraditionalForm]],ExpressionUUID->"a25b6cd4-9358-45a2-82ba-7376725d88f6"], " \[RightArrow] x=x\[CloseCurlyQuote] mod n (and so ", Cell[BoxData[ FormBox[ SubscriptBox["e", RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}]], TraditionalForm]],ExpressionUUID-> "de266a8e-6792-44a4-a5c1-1d3ead53414c"], " is injective, and so since |X|=|Y| it is also bijective)\n\nAt this point \ we now that indeed this is cryptosystem, so it remains to find the inverse\n\n\ ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", 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CellLabel->"In[22]:=",ExpressionUUID->"b5528e8a-ba34-40a1-83ff-291c0bab12fd"], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{{3.823854471940379*^9, 3.8238544719424763`*^9}},ExpressionUUID->"310b63d5-edf4-473f-a009-\ 2b012bcab709"], Cell[TextData[{ "Note that the number of keys of the affine cs is just n\[Cross]\[Phi](n) ", Cell[BoxData[ FormBox[ RowBox[{"<", SuperscriptBox["n", "2"]}], TraditionalForm]],ExpressionUUID-> "0012326e-13b6-465d-a5db-6bb3e2bdbef4"], " and so this quite prone to brute force attacks! 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Substitution Cryptosystem \[ScriptCapitalS]=", StyleBox["(X,Y,K,e,d)\[LineSeparator]", "DisplayFormula"], "\n\[Bullet] X=Y=", Cell[BoxData[ FormBox[ SubscriptBox["\[DoubleStruckCapitalZ]", "n"], TraditionalForm]], ExpressionUUID->"a5b28c48-f70d-489d-a3f8-05bb66fcf585"], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Bullet]", " ", "K"}], " ", "=", RowBox[{ SubscriptBox["\[DoubleStruckCapitalP]", "n"], "=", RowBox[{"{", RowBox[{"\[Gamma]", ":", RowBox[{ "\[Gamma]", " ", "is", " ", "a", " ", "permutation", " ", "in", " "}]}]}]}]}], TraditionalForm]],ExpressionUUID-> "71a67e43-6b5f-4591-850a-55a8b74a3aa4"], Cell[BoxData[ FormBox[ SubscriptBox["\[DoubleStruckCapitalZ]", "n"], TraditionalForm]], ExpressionUUID->"0c2e5e8a-5ed9-463e-ada4-6a0843804475"], "}\n\[Bullet] ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["e", "\[Gamma]"], "(", "x", ")"}], "=", RowBox[{"\[Gamma]", "(", "x", ")"}]}], TraditionalForm]],ExpressionUUID-> "3579334f-31c7-4e2c-8a26-6b8cd39f03cb"], " \n\[Bullet] 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However, \ frequencies of pairs of letters and other sequences will give statistics and \ evidence as well.\ \>", "Text", CellChangeTimes->{{3.7282123164298573`*^9, 3.728212407893959*^9}, { 3.7282124668862963`*^9, 3.72821252969088*^9}, {3.728213421210248*^9, 3.728213436649889*^9}},ExpressionUUID->"dc4f87e1-cc8f-4e60-9b49-\ 352194ee70e1"] }, Closed]], Cell[CellGroupData[{ Cell["Vigin\[EGrave]re cryptosystem and one-time pad", "Subsubsection", CellChangeTimes->{{3.728194881327486*^9, 3.72819489643839*^9}, { 3.728204510439024*^9, 3.7282045605558987`*^9}, {3.728212600001322*^9, 3.728212601200492*^9}},ExpressionUUID->"d5c62ff9-c9d6-40c2-a4e6-\ 3f351f7e2918"], Cell["\<\ Vigin\[EGrave]re came up with a system that seems to defeat basic frequency \ analysis.\ \>", "Text", CellChangeTimes->{{3.728213438692704*^9, 3.728213467585339*^9}},ExpressionUUID->"7613d069-7408-472b-aadc-\ 2e0fd65fe107"], Cell[TextData[{ StyleBox["Definition 5. 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The idea is to first discover the key length. That can be \ done by noticing that very frequent letters or sequences of letters will \ match in the ciphertext in multiples of the key length. Then one can compute \ the gcd between those lengths and get the size of the key. When the key size \ is known,it is enough to perform frequency analysis for each segment |m|/|k|!\ \ \>", "Text", CellChangeTimes->{{3.728215810721459*^9, 3.728215988636177*^9}},ExpressionUUID->"2e63281c-9e8a-4fd2-8940-\ 57e3530cb254"], Cell[TextData[{ StyleBox["Example. ", FontWeight->"Bold"], "Consider a simple key and assume the word \[OpenCurlyDoubleQuote]the\ \[CloseCurlyDoubleQuote] should appear often." }], "Text", CellChangeTimes->{{3.728216050915916*^9, 3.728216088352832*^9}},ExpressionUUID->"5734149e-cf71-4649-a182-\ b2d56e403bf4"], Cell[BoxData[ RowBox[{ RowBox[{"k", "=", RowBox[{"tonumber", "[", "\"\\"", "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.728216033012891*^9, 3.728216034491456*^9}, { 3.728216238557486*^9, 3.728216238636457*^9}}, CellLabel->"In[84]:=",ExpressionUUID->"03c00702-b1c8-4051-9db7-3373679fabab"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"tostring", "[", RowBox[{"enc", "[", RowBox[{"k", ",", RowBox[{"tonumber", "[", "m", "]"}]}], "]"}], "]"}]], "Input", CellChangeTimes->{3.728216045620261*^9}, CellLabel->"In[85]:=",ExpressionUUID->"71d7892b-56a2-4a28-a3df-99fe60fd74a0"], Cell[BoxData["\<\"\ tieqauhpfuhfrjgitfovsnaoitbfsftpnblmsjdfscyuhfioirujtjetogtietemfjsiaoduhftzrb\ noypffvjlneobmetsfdjsiexhpiotieoanepfdhbrjtzaodhopdximlthfpiesdttiexebkuhsovgi\ tiewamlfypfeaskoetsgoshfittsumyiitbsouhfrtkfeqesaoduhffjneesoglpsuciimdseoaodj\ wjlmsurjkfdpwouqootiefwjtigsebtweogfaocfaodgusiputaogfruhpsfwioxovleautfmqtuoq\ ojspnbnedfsurpynycrptiessbneypuximllnpwnyoanejsuhflprewieoimazmzvfnhebndevppnu\ hf\"\>"], "Output", CellChangeTimes->{3.823856754637506*^9}, CellLabel->"Out[85]=",ExpressionUUID->"d05bb8bb-ffd8-4e93-ab0d-5b196de2b64d"] }, Open ]], Cell[TextData[{ "\n\ntieqauhpf", StyleBox["uh", FontWeight->"Bold"], "frjgitfovsnaoitbfsftpnblmsjdfscy", StyleBox["uh", FontWeight->"Bold"], "fioirujtjetogtietemfjsiaod", StyleBox["uh", FontWeight->"Bold"], "ftzrbnoypffvjlneobmetsfdjsiexhpiotieoanepfdhbrjtzaodhopdximlthfpiesdttiexebk\ ", StyleBox["uh", FontWeight->"Bold"], "sovgitiewamlfypfeaskoetsgoshfittsumyiitbso", StyleBox["uh", FontWeight->"Bold"], "frtkfeqesaod", StyleBox["uh", FontWeight->"Bold"], "ffjneesoglpsuciimdseoaodjwjlmsurjkfdpwouqootiefwjtigsebtweogfaocfaodgusiputa\ ogfruhpsfwioxovleautfmqtuoqojspnbnedfsurpynycrptiessbneypuximllnpwnyoanejsuhfl\ prewieoimazmzvfnhebndevppnuhf" }], "Text", CellChangeTimes->{{3.8238567959223547`*^9, 3.823856801064025*^9}},ExpressionUUID->"43d27e2f-94a8-4dbd-9c88-\ 8704ab07e481"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"StringLength", "[", "\"\<\!\(\*StyleBox[\(\*StyleBox[\"uh\",FontWeight->\"Bold\"]frtkfeqesaod\)]\ \)\>\"", "]"}]], "Input", CellChangeTimes->{{3.8238569043149843`*^9, 3.823856915580469*^9}}, CellLabel->"In[86]:=",ExpressionUUID->"fb7893ad-ad63-4149-9018-7b3cad63700d"], Cell[BoxData["62"], "Output", CellChangeTimes->{3.823856915919509*^9}, CellLabel->"Out[86]=",ExpressionUUID->"a8fd0491-b778-4ceb-9c12-2dc1957a0386"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"StringLength", "[", "\"\<\!\(\*StyleBox[\(\*StyleBox[\"uh\",FontWeight->\"Bold\"]\ frjgitfovsnaoitbfsftpnblmsjdfscy\)]\)\>\"", "]"}]], "Input", 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Vernam (one-time pad) Cryptosystem \[ScriptCapitalV]=", StyleBox["(X,Y,K,e,d)", "DisplayFormula"], "\n\[Bullet] X=Y=K=", Cell[BoxData[ FormBox[ SubsuperscriptBox["\[DoubleStruckCapitalZ]", "n", "d"], TraditionalForm]], ExpressionUUID->"9fda9d5b-06fb-4178-a3d3-c6ecc9e2e287"], " \n\[Bullet] ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["e", "k"], "(", "x", ")"}], "=", " ", RowBox[{"x", "+", " ", "k"}]}], TraditionalForm]],ExpressionUUID-> "d1784ef6-0283-4869-bcbd-9aa1c1e71543"], "\n\[Bullet] ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["d", "k"], "(", "y", ")"}], "=", " ", RowBox[{"y", "-", " ", "k"}]}], TraditionalForm]],ExpressionUUID-> "99b8893a-0aff-41cf-80d1-a2c742364c72"], "\n\nThe key is picked randomly with uniform distribution and can only be \ used once and has the length of the message, that is d is plaintext length." }], "Text", CellChangeTimes->{{3.72821373731842*^9, 3.7282138155645533`*^9}, { 3.728219487569324*^9, 3.7282195316385736`*^9}, {3.728219621187593*^9, 3.728219624515744*^9}, {3.823857182883363*^9, 3.823857212699307*^9}, { 3.823858055603079*^9, 3.823858070956958*^9}},ExpressionUUID->"30a559b2-3970-4853-8a03-\ 9be2cde15956"] }, Closed]], Cell[CellGroupData[{ Cell["Shannon theorem", "Subsubsection", CellChangeTimes->{{3.728194881327486*^9, 3.72819489643839*^9}, { 3.728204510439024*^9, 3.728204566995632*^9}, {3.8244665748875637`*^9, 3.824466574951733*^9}},ExpressionUUID->"65a03754-f439-4b24-9798-\ 875110969274"], Cell["\<\ Indeed Shannon showed that the one-time pad was unbreakable. He realized that \ one could always guess the key and attack in this way. If this was the best \ one could do, then the system was perfect. \ \>", "Text", CellChangeTimes->{{3.7282196420513153`*^9, 3.72821970764828*^9}, { 3.7282240947028933`*^9, 3.72822410495684*^9}},ExpressionUUID->"127f324c-688f-47b3-9e12-\ 45c6455e2c0d"], Cell[TextData[{ StyleBox["Assumption", FontWeight->"Bold"], ". From this point on we assume X and K are independent random variables, \ that is P(X=x \[Wedge] K=k)=P(X=x)P(K=k) (independence means)\nWe assume \ P(X=x)>0 and P(Y=y)>0 (no meaningless plaintexts or cyphertexts)" }], "Text", CellChangeTimes->{{3.728219709265279*^9, 3.728219737662929*^9}, { 3.728219791278351*^9, 3.728219826459852*^9}, {3.728219939088841*^9, 3.728219994485814*^9}, {3.823857479281315*^9, 3.8238574888413763`*^9}, { 3.8238575420252323`*^9, 3.823857554729128*^9}, {3.8242069284636*^9, 3.82420694784025*^9}, {3.8242069950563383`*^9, 3.824206999831778*^9}},ExpressionUUID->"8d07e80c-42f0-4a3a-80a8-\ 57bf838d74c1"], Cell[TextData[{ "We can define P(Y=y)=", Cell[BoxData[{ FormBox[ RowBox[{ SubscriptBox["\[Sum]", RowBox[{"x", "\[Element]", "X"}]], RowBox[{"P", "(", RowBox[{"Y", "=", RowBox[{ RowBox[{"y", "\[Wedge]", "X"}], "=", "x"}]}], ")"}]}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{"=", RowBox[{ SubscriptBox["\[Sum]", RowBox[{"x", "\[Element]", "X"}]], RowBox[{ RowBox[{"P", "(", RowBox[{"Y", "=", RowBox[{ RowBox[{"y", "|", "X"}], "=", "x"}]}], ")"}], RowBox[{"P", "(", RowBox[{"X", "=", "x"}], ")"}]}]}]}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{"=", RowBox[{ SubscriptBox["\[Sum]", RowBox[{"x", "\[Element]", "X"}]], RowBox[{ RowBox[{"P", "(", RowBox[{"K", "\[Element]", RowBox[{"{", RowBox[{ RowBox[{"k", ":", RowBox[{ SubscriptBox["e", "k"], "(", "x", ")"}]}], "=", "y"}], "}"}]}], ")"}], RowBox[{"P", "(", RowBox[{"X", "=", "x"}], ")"}], " ", RowBox[{"(*", ")"}]}]}]}], TraditionalForm]}],ExpressionUUID-> "b9bff96e-8824-4363-a054-3d4707e0faed"], " \n\nP(Y=y|X=x)=", Cell[BoxData[ FormBox[ RowBox[{"P", "(", RowBox[{"K", "\[Element]", RowBox[{"{", RowBox[{ RowBox[{"k", ":", RowBox[{ SubscriptBox["e", "k"], "(", "x", ")"}]}], "=", "y"}], "}"}]}]}], TraditionalForm]],ExpressionUUID->"f9f82049-24aa-4912-a4f7-c597f70484be"], ") (**)" }], "Text", CellChangeTimes->{{3.728219999510445*^9, 3.728220127140478*^9}, { 3.7282990336802444`*^9, 3.728299048274769*^9}, {3.791198583868801*^9, 3.791198660232643*^9}, {3.823857661280505*^9, 3.8238576783421993`*^9}, 3.8242070936914663`*^9, {3.8242077035366983`*^9, 3.824207722723627*^9}, { 3.824208175496524*^9, 3.824208177597374*^9}},ExpressionUUID->"9cdb1775-6ca5-4b05-ae35-\ 519002d6b838"], Cell[TextData[{ StyleBox["Definition 8[by Shannon]. ", FontWeight->"Bold"], "A cryptosystem is said to be ", StyleBox["perfectly secure", FontWeight->"Bold"], " if \n\t\t\t\tP(X=x|Y=y)=P(X=x) (which means X is independent Y)\n\n", StyleBox["Theorem 9. ", FontWeight->"Bold"], "Let (X,Y,K,e,d) be a cryptosystem such that |X|=|Y|=|K|. \nP(Y=y)>0 for all \ y (0)\n\nThen, the system is perfectly secure iff \n1) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["\[ForAll]", RowBox[{"x", ",", "y"}]], RowBox[{ SubsuperscriptBox["\[Exists]", "k", "1"], RowBox[{ SubscriptBox["e", "k"], "(", "x", ")"}]}]}], "=", "y"}], TraditionalForm]],ExpressionUUID->"6789d9a8-2434-4ac9-8e5b-62c40d659e2a"], " and (let us call the key ", Cell[BoxData[ FormBox[ SubscriptBox["k", "xy"], TraditionalForm]],ExpressionUUID-> "a4e3823e-7456-40ac-8551-2d22ef5664f8"], ")\n2) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", RowBox[{"(", RowBox[{"K", "=", "k"}], ")"}]}], "=", FractionBox["1", RowBox[{"|", "K", "|"}]]}], TraditionalForm]],ExpressionUUID-> "b87b3acc-4ca3-4e0a-8f77-6693fea2e0d8"], "." }], "Text", CellChangeTimes->{{3.728220154969623*^9, 3.728220392583291*^9}, { 3.728220460100967*^9, 3.7282204614930363`*^9}, {3.7282991671716046`*^9, 3.7282991708280497`*^9}, {3.791196357343309*^9, 3.791196359998735*^9}, { 3.791196612303162*^9, 3.791196620254648*^9}, {3.8238577310347633`*^9, 3.823857736340427*^9}, {3.8242070595040894`*^9, 3.8242070659442253`*^9}, { 3.8242071374402*^9, 3.824207152833487*^9}, {3.824207587152371*^9, 3.8242076025608683`*^9}, {3.824208222393281*^9, 3.824208224429331*^9}},ExpressionUUID->"02d245e3-8fea-49b8-8419-\ f9509a5ef585"], Cell[TextData[{ "Let us do the proof:\n\n\[DoubleLeftArrow]) \nP(X=x|Y=y) ", Cell[BoxData[ FormBox[ RowBox[{"=", RowBox[{ FractionBox[ RowBox[{"P", "(", RowBox[{"X", "=", RowBox[{ RowBox[{"x", "\[Wedge]", "Y"}], "=", "y"}]}], ")"}], RowBox[{"P", "(", RowBox[{"Y", "=", "y"}], ")"}]], "=", FractionBox[ RowBox[{"P", "(", RowBox[{"X", "=", RowBox[{ RowBox[{"x", "\[Wedge]", "Y"}], "=", "y"}]}], ")"}], RowBox[{"P", "(", RowBox[{"Y", "=", "y"}], ")"}]]}]}], TraditionalForm]],ExpressionUUID-> "eff6b495-1f9e-4dfa-ba1a-ffc3c310dd0d"], " = By (*)= ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{"P", "(", RowBox[{ RowBox[{"K", "\[Element]", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"k", ":", RowBox[{ SubscriptBox["e", "k"], "(", "x", ")"}]}], "=", "y"}], "}"}], "\[Wedge]", "X"}]}], "=", "x"}], ")"}], RowBox[{ SubscriptBox["\[Sum]", RowBox[{ RowBox[{"x", "'"}], "\[Element]", "X"}]], RowBox[{ RowBox[{"P", "(", RowBox[{"K", "\[Element]", RowBox[{"{", RowBox[{ RowBox[{"k", ":", RowBox[{ SubscriptBox["e", "k"], "(", RowBox[{"x", "'"}], ")"}]}], "=", "y"}], "}"}]}], ")"}], RowBox[{"P", "(", RowBox[{"X", "=", RowBox[{"x", "'"}]}], ")"}]}]}]], "=", " ", RowBox[{ RowBox[{"by", " ", "1"}], " ", "=", " "}]}], TraditionalForm]], ExpressionUUID->"675766ab-0aea-44cc-a4d6-961e42b21507"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ RowBox[{"P", "(", RowBox[{"K", "=", SubscriptBox["k", "xy"]}], ")"}], RowBox[{"P", "(", RowBox[{"X", "=", "x"}], ")"}]}], RowBox[{ SubscriptBox["\[Sum]", RowBox[{ RowBox[{"x", "'"}], "\[Element]", "X"}]], RowBox[{ RowBox[{"P", "(", RowBox[{"K", "=", SubscriptBox["k", RowBox[{ RowBox[{"x", "'"}], "y"}]]}], ")"}], RowBox[{"P", "(", RowBox[{"X", "=", RowBox[{"x", "'"}]}], ")"}]}]}]], TraditionalForm]],ExpressionUUID-> "ee270838-fc70-4142-8896-218d5e692abc"], " =\nby 2 = ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ FractionBox["1", RowBox[{"|", "K", "|"}]], RowBox[{"P", "(", RowBox[{"X", "=", "x"}], ")"}]}], RowBox[{ SubscriptBox["\[Sum]", RowBox[{ RowBox[{"x", "'"}], "\[Element]", "X"}]], RowBox[{ FractionBox["1", RowBox[{"|", "K", "|"}]], RowBox[{"P", "(", RowBox[{"X", "=", RowBox[{"x", "'"}]}], ")"}]}]}]], TraditionalForm]],ExpressionUUID-> "40f18d63-8fc1-4174-a147-31a071eafdfe"], "=", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{"P", "(", RowBox[{"X", "=", "x"}], ")"}], RowBox[{ SubscriptBox["\[Sum]", RowBox[{ RowBox[{"x", "'"}], "\[Element]", "X"}]], RowBox[{"P", "(", RowBox[{"X", "=", RowBox[{"x", "'"}]}], ")"}]}]], "="}], TraditionalForm]], ExpressionUUID->"672b5447-8e80-4803-a7d3-e95045dc16af"], Cell[BoxData[ FormBox[ RowBox[{"P", "(", RowBox[{"X", "=", "x"}], ")"}], TraditionalForm]],ExpressionUUID-> "5f947ee3-65d3-4d7f-8432-a90a9f8f0b6e"], ". So the system is perfectly secure!\n\n\[DoubleRightArrow]) we have that \ Y is independent of X (the assumption as the system is perfectly secure)\n \ Fix an x and for any y, ", Cell[BoxData[ FormBox[ RowBox[{"P", "(", RowBox[{"K", "\[Element]", RowBox[{"{", RowBox[{ RowBox[{"k", ":", RowBox[{ SubscriptBox["e", "k"], "(", "x", ")"}]}], "=", "y"}], "}"}]}], ")"}], TraditionalForm]],ExpressionUUID-> "8d43b3db-6c05-4f27-b82a-b2e7a84cd928"], "=P(Y=y|X=x)=P(Y=y)>0 by assumption (0)\n\nThen, for fixed x, and any y ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ RowBox[{"k", ":", RowBox[{ SubscriptBox["e", "k"], "(", "x", ")"}]}], "=", "y"}], "}"}], TraditionalForm]],ExpressionUUID->"907707b6-c8a2-4c34-8be1-908270a33aba"], " is not empty (otherwise it would have probability 0)\n\nconsider the \ following map\n", Cell[BoxData[ FormBox[ SubscriptBox["f", "x"], TraditionalForm]],ExpressionUUID-> "bc7ef5c6-d882-4730-ab72-6eb5bb184ce5"], ":K\[RightArrow]Y where ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["f", "x"], "(", "k", ")"}], "=", RowBox[{ SubscriptBox["e", "k"], "(", "x", ")"}]}], TraditionalForm]], ExpressionUUID->"98756a5f-2159-4aaa-8f61-461917364322"], ", so ", Cell[BoxData[ FormBox[ SubscriptBox["f", "x"], TraditionalForm]],ExpressionUUID-> "447a99ee-e17d-44da-b6a9-d8939a99097b"], " is surjective (as for any y there exists and k such that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["f", "x"], "(", "k", ")"}], "=", RowBox[{ SubscriptBox["e", "k"], "(", "x", ")"}]}], TraditionalForm]], ExpressionUUID->"8a5d35f5-2969-431e-b1b3-646d56b33625"], "=y), which means that , since |X|=|Y|=|K|, that it is bijective, and so \n\ 1) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[ForAll]", RowBox[{"x", ",", "y"}]], RowBox[{ SubsuperscriptBox["\[Exists]", "k", "1"], RowBox[{ SubscriptBox["e", "k"], "(", "x", ")"}]}]}], " ", "=", " ", RowBox[{"y", " ", "and", " ", "moreover"}]}], ",", " ", RowBox[{"k", "=", RowBox[{ RowBox[{ SubsuperscriptBox["f", "x", RowBox[{"-", "1"}]], "(", "y", ")"}], " ", "=", RowBox[{ SubscriptBox["k", "xy"], "which", " ", "is", " ", "unique"}]}]}], ","}], TraditionalForm]],ExpressionUUID-> "26cbd509-fe0c-45fb-8150-7930b6ac0932"], " since f is a bijection.\n\nNow, observe that ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ RowBox[{"P", "(", RowBox[{"K", "=", SubscriptBox["k", "xy"]}], ")"}], TraditionalForm], "=", " ", RowBox[{ FormBox[ RowBox[{"P", "(", RowBox[{"K", "\[Element]", RowBox[{"{", RowBox[{ RowBox[{"k", ":", RowBox[{ SubscriptBox["e", "k"], "(", "x", ")"}]}], "=", "y"}], "}"}]}], ")"}], TraditionalForm], "=", RowBox[{"P", "(", RowBox[{"Y", "=", RowBox[{ RowBox[{"y", "|", "X"}], "=", "x"}]}], ")"}]}]}], TraditionalForm]], ExpressionUUID->"897bc175-0268-48e6-959a-11d9714aee58"], "=P(Y=y) (Because the system is perfectly secure)\n\n", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ RowBox[{"P", "(", RowBox[{"K", "=", SubscriptBox["k", "xy"]}], ")"}], TraditionalForm], "=", " ", RowBox[{ FormBox[ RowBox[{"P", "(", RowBox[{"K", "\[Element]", RowBox[{"{", RowBox[{ RowBox[{"k", ":", RowBox[{ SubscriptBox["e", "k"], "(", "x", ")"}]}], "=", "y"}], "}"}]}], ")"}], TraditionalForm], "=", RowBox[{"P", "(", RowBox[{"Y", "=", RowBox[{ RowBox[{"y", "|", "X"}], "=", "x"}]}], ")"}]}]}], TraditionalForm]], ExpressionUUID->"d7462d02-4bcf-4c52-ba8e-efa3cfb43090"], "=P(Y=y)\n", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ RowBox[{"P", "(", RowBox[{"K", "=", SubscriptBox["k", RowBox[{ RowBox[{"x", "'"}], "y"}]]}], ")"}], TraditionalForm], "=", " ", RowBox[{ FormBox[ RowBox[{"P", "(", RowBox[{"K", "\[Element]", RowBox[{"{", RowBox[{ RowBox[{"k", ":", RowBox[{ SubscriptBox["e", "k"], "(", "x", ")"}]}], "=", "y"}], "}"}]}], ")"}], TraditionalForm], "=", RowBox[{"P", "(", RowBox[{"Y", "=", RowBox[{ RowBox[{"y", "|", "X"}], "=", RowBox[{"x", "'"}]}]}], ")"}]}]}], TraditionalForm]],ExpressionUUID-> "95d8a244-f52e-4376-9419-d2613e7d27cb"], "=P(Y=y)\n\nAnd so ", Cell[BoxData[ FormBox[ RowBox[{"P", "(", RowBox[{"K", "=", SubscriptBox["k", "xy"]}], ")"}], TraditionalForm]],ExpressionUUID-> "6f3573fc-4ae1-4076-b4e7-daab285896d9"], "=", Cell[BoxData[ FormBox[ RowBox[{"P", "(", RowBox[{"K", "=", SubscriptBox["k", RowBox[{ RowBox[{"x", "'"}], "y"}]]}], ")"}], TraditionalForm]],ExpressionUUID-> "0f3ce608-900b-4fe1-8a5e-d2a360a65732"], " for any x and x\[CloseCurlyQuote] (2) . Now consider\n", Cell[BoxData[ FormBox[ SubscriptBox["g", "y"], TraditionalForm]],ExpressionUUID-> "9ada97f3-18e9-490a-9177-082a62d6b84d"], ":X\[RightArrow]K where ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["g", "y"], "(", "x", ")"}], "=", SubscriptBox["k", "xy"]}], TraditionalForm]],ExpressionUUID-> "37895868-3a3f-4704-b731-f39813c14021"], " again, this function is injective if ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["g", "y"], "(", "x", ")"}], TraditionalForm]],ExpressionUUID-> "562fab78-49ea-4528-8b10-836d55fe1b1e"], "=", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["g", "y"], "(", RowBox[{"x", "'"}], ")"}], TraditionalForm]],ExpressionUUID-> "86070a5a-19bb-4d5f-b183-dfc6af85df77"], " then ", Cell[BoxData[ FormBox[ SubscriptBox["k", "xy"], TraditionalForm]],ExpressionUUID-> "b61242e5-ee29-4922-97b7-959aea75e014"], "=", Cell[BoxData[ FormBox[ SubscriptBox["k", RowBox[{ RowBox[{"x", "'"}], "y"}]], TraditionalForm]],ExpressionUUID-> "a3b4190a-824a-4de0-93a6-65f566cdee0a"], " which imply x=x\[CloseCurlyQuote], and so it is bijective and so\n\n", Cell[BoxData[ FormBox[ RowBox[{"P", "(", RowBox[{"K", "=", SubscriptBox["k", "xy"]}], ")"}], TraditionalForm]],ExpressionUUID-> "4346cc6d-4294-417f-bc1d-825999378ff3"], "=", Cell[BoxData[ FormBox[ RowBox[{"constant", " "}], TraditionalForm]],ExpressionUUID-> "8ba533e3-6fad-4da4-a83a-7cfb0405ae7b"], "(not depending on x) for any k and given y there exists x such that ", Cell[BoxData[ FormBox[ SubscriptBox["k", "xy"], TraditionalForm]],ExpressionUUID-> "e6aa3e4c-4044-4879-8775-51d1574084dc"], "=k\nand so P(K=k) is uniform which implies P(K=k)=1/|K|. QED" }], "Text", CellChangeTimes->{{3.824207395628428*^9, 3.824207470749221*^9}, { 3.824207501208251*^9, 3.824207579459744*^9}, {3.824207626986332*^9, 3.824207965470539*^9}, {3.8242080846543303`*^9, 3.8242084309730453`*^9}, { 3.8242084686060963`*^9, 3.824208481576435*^9}, {3.824208615116453*^9, 3.824208826739008*^9}, {3.824208860552608*^9, 3.824209289235338*^9}},ExpressionUUID->"4a0e0977-8ed2-4f4f-a433-\ 08aad3c8ab1d"], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{{3.824208066448986*^9, 3.824208082847793*^9}},ExpressionUUID->"b33db730-6784-4d0d-bdc3-\ 7de582edab1a"], Cell[TextData[{ "Note that Vernam cryptosystem fulfills condition 1) and 2), and thus it is \ perfectly secure.\n\nThe problem with the One-time pad is ", StyleBox["how to store and renew", FontWeight->"Bold"], " the key!" }], "Text", CellChangeTimes->{{3.728220397357753*^9, 3.7282204508769207`*^9}, { 3.824209808289147*^9, 3.824209824374707*^9}},ExpressionUUID->"288890c0-bff7-430e-9662-\ 5ab1318afc72"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Modern cryptosystems", "Section", CellChangeTimes->{{3.728062293677641*^9, 3.728062304019003*^9}, { 3.728193259258326*^9, 3.728193262544937*^9}, {3.728289540604205*^9, 3.7282895716818933`*^9}},ExpressionUUID->"da291f0b-a1cf-4db6-ae1e-\ 3c65ef67b1f3"], Cell[CellGroupData[{ Cell["Hill cryptosystem", "Subsubsection", CellChangeTimes->{{3.728194881327486*^9, 3.72819489643839*^9}, { 3.728204510439024*^9, 3.728204566995632*^9}, {3.728220468101565*^9, 3.728220486139888*^9}},ExpressionUUID->"0c878685-2045-4279-bb8c-\ a8d42c36bd7f"], Cell[TextData[{ "Before finishing this section we shall look into an interesting \ cryptosystem that can be easily attacked\n\n", StyleBox["Definition 6", FontWeight->"Bold"], ". Hill Cryptosystem \[ScriptCapitalH]=", StyleBox["(X,Y,K,e,d)", "DisplayFormula"], "\n\[Bullet] X=Y=", Cell[BoxData[ FormBox[ SubsuperscriptBox["\[DoubleStruckCapitalZ]", "n", "m"], TraditionalForm]], ExpressionUUID->"451c1d5e-59f7-406d-b8ec-4b6082ea1bd6"], "\n\[Bullet] K= GL(m,", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["\[DoubleStruckCapitalZ]", "n"], ")"}], " "}], TraditionalForm]],ExpressionUUID->"22e61536-2bb3-4340-a5fd-ffac091ee5a6"], "called the general linear group, that is, the set of all invertible m\ \[Cross]m matrices modulo n;\n\[Bullet] ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["e", "k"], "(", "x", ")"}], "=", " ", RowBox[{"k", " ", "x"}]}], TraditionalForm]],ExpressionUUID-> "91da603d-b95c-482d-bc45-094ee9e78752"], "\n\[Bullet] ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["d", "k"], "(", "x", ")"}], "=", " ", RowBox[{ SuperscriptBox["k", RowBox[{"-", "1"}]], " ", "x"}]}], TraditionalForm]],ExpressionUUID-> "a11499b9-503b-4c27-a939-9d837944b730"] }], "Text", CellChangeTimes->{{3.728220514089445*^9, 3.728220708951079*^9}, { 3.728220747347621*^9, 3.728220776721796*^9}, {3.728328628092615*^9, 3.728328632369574*^9}, {3.728655807371352*^9, 3.728655809089116*^9}, { 3.824209672875005*^9, 3.8242096785007267`*^9}},ExpressionUUID->"424494ed-5a62-40f2-8bb2-\ 88e922472a5f"], Cell[TextData[{ "Hill implemented mechanically the cipher for n=26 and m=6 in 1929, and it \ was the first cipher that could operate on more than three symbols at once. \ However,", StyleBox[" his machine did not sell", FontWeight->"Bold"], "!\n\nAs we shall see, for n=26=13\[Cross]2, thanks to the", StyleBox[" Chinese Remainder Theorem,", FontWeight->"Bold"], " a matrix A is is invertible iff its determinant is invertible modulo 13 \ and modulo 2. The order of GL(n, q) when q is prime\n\n|GL(k, q)|= ", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Product]", RowBox[{"i", "=", "0"}], RowBox[{"m", "-", "1"}]], RowBox[{"(", RowBox[{ SuperscriptBox["q", "m"], "-", SuperscriptBox["q", "i"]}], ")"}]}], TraditionalForm]],ExpressionUUID-> "a0b2154d-9814-44c0-8493-c55faa97f2f6"], "\n\nThe first column can be any but the zero vector; \nThe second column \ can be anything but multiples of the first column; \nThe kth column can be \ any vector not in the linear span of the first k \[Minus] 1 columns.\n\nThus, \ |GL(k,", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[DoubleStruckCapitalZ]", "26"], ")"}], "|"}], " ", "=", RowBox[{ SubsuperscriptBox["\[Product]", RowBox[{"i", "=", "0"}], RowBox[{"k", "-", "1"}]], RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox["2", "k"], "-", SuperscriptBox["2", "i"]}], ")"}], RowBox[{ SubsuperscriptBox["\[Product]", RowBox[{"i", "=", "0"}], RowBox[{"k", "-", "1"}]], RowBox[{"(", RowBox[{ SuperscriptBox["13", "k"], "-", SuperscriptBox["13", "i"]}], ")"}]}]}]}]}], TraditionalForm]], ExpressionUUID->"f86e9396-b044-45bd-a1fb-0d5209f9d48c"], ". 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willstrikedownupontheewithgreatvengeanceandfuriousangerthosewhowouldattempttop\ oisonanddestroymybrothersandyouwillknowmynameisthelordwhenilaymyvengeanceupont\ hee\>\"", "\[IndentingNewLine]"}]], "Input", CellChangeTimes->{{3.824210191354327*^9, 3.82421020113519*^9}},ExpressionUUID->"5acd478e-8f97-4606-b1a1-\ 670d54e08239"], Cell["\<\ This cryptosystem is going to avoid frequency analysis very well...\ \>", "Text", CellChangeTimes->{{3.824210202428417*^9, 3.8242102162142487`*^9}},ExpressionUUID->"ae9164cd-f9b0-4fbb-88b2-\ 7b65abd22cce"] }, Closed]], Cell[CellGroupData[{ Cell["Cryptoanalysis of the Hill cryptosystem", "Subsubsection", CellChangeTimes->{{3.7283141446982813`*^9, 3.728314161580213*^9}, { 3.824466453835608*^9, 3.824466459811576*^9}},ExpressionUUID->"ab7fdf82-84f3-49af-92cb-\ 5553a3bb8161"], Cell[TextData[{ "Until now we have assumed that the ", StyleBox["attacker has no knowledge about the plaintext", FontWeight->"Bold"], ", but in general this is the case. Hill cipher is ", StyleBox["trivially attacked if the attacker has some knowledge about the \ plaintexts", FontWeight->"Bold"], ". In cryptography four types of attacks are considered:\n\n\[Bullet] ", StyleBox["Ciphertext only attack: ", FontWeight->"Bold"], "The attacker only knows the ciphertext\[LineSeparator]\n\[Bullet] ", StyleBox["Know plaintext attack: ", FontWeight->"Bold"], "The attacker knows some plaintexts and the corresponding ciphertext .\n\n\ \[Bullet] ", StyleBox["Chosen plaintext attack: ", FontWeight->"Bold"], "The attacker can choose some plaintexts for which he obtains corresponding \ ciphertexts (Eve had access to the encryption function).\n\n\[Bullet] ", StyleBox["Chosen ciphertext attack: ", FontWeight->"Bold"], "The attacker can choose some ciphertext for which he obtains corresponding \ plaintexts (Eve had access to the decryption function).\n\nThe previous \ ciphers were broken using ", StyleBox["ciphertext only attack", FontSlant->"Italic"], ", ", StyleBox["Hill cipher can be easily broken", FontWeight->"Bold"], " with the ", StyleBox["know plaintext attack", FontSlant->"Italic"], ", or any of the others. \n\nTo realize this, notice that we just have to \ know the ciphertext of m independent plaintexts and we are able to extract \ the key by solving a system of linear equations (modulo 26)!\n\nWe exemplify \ the procedure for m=3;" }], "Text", CellChangeTimes->{{3.728314164323268*^9, 3.728314522112159*^9}, { 3.728328362234589*^9, 3.728328364279254*^9}, {3.728328572199069*^9, 3.7283286150293407`*^9}, {3.728328675564786*^9, 3.728328716693007*^9}, { 3.728382968788871*^9, 3.728383000770999*^9}, {3.728541673672325*^9, 3.728541687633692*^9}, {3.7285422826420307`*^9, 3.728542286330962*^9}, { 3.728542855543461*^9, 3.728542886136156*^9}, {3.792152785192519*^9, 3.7921527960486603`*^9}, 3.8242103717777023`*^9, {3.824210402583179*^9, 3.824210406453071*^9}, {3.824210482096756*^9, 3.824210482853414*^9}},ExpressionUUID->"5be02816-d7f4-462f-afab-\ 5b8bd6f31056"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"kx", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"k11", ",", "k12", ",", "k13"}], "}"}], ",", RowBox[{"{", RowBox[{"k21", ",", "k22", ",", "k23"}], "}"}], ",", RowBox[{"{", RowBox[{"k31", ",", "k32", ",", "k33"}], "}"}]}], "}"}]}]], "Input", CellChangeTimes->{{3.728541689717079*^9, 3.728541739504319*^9}, { 3.728541845407303*^9, 3.7285418476024*^9}}, CellLabel->"In[28]:=",ExpressionUUID->"9cd2dc33-27f4-4a85-872a-399a76f5fa79"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"k11", ",", "k12", ",", "k13"}], "}"}], ",", RowBox[{"{", RowBox[{"k21", ",", "k22", ",", "k23"}], "}"}], ",", RowBox[{"{", RowBox[{"k31", ",", "k32", ",", "k33"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.824210485131391*^9}, CellLabel->"Out[28]=",ExpressionUUID->"d265b578-351c-4eee-bf76-812d22f8dfa9"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", "kx", "]"}]], "Input", CellChangeTimes->{{3.728656420900897*^9, 3.728656425002069*^9}}, CellLabel->"In[29]:=",ExpressionUUID->"2966e8af-7fbf-4732-af56-f9ff2d3e3cd4"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"k11", "k12", "k13"}, {"k21", "k22", "k23"}, {"k31", "k32", "k33"} }, GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.728656425243393*^9, 3.793344393925357*^9, 3.8242104870069313`*^9}, CellLabel-> "Out[29]//MatrixForm=",ExpressionUUID->"fdb798d4-1e6d-482c-a8c7-\ 3e68df9b7e84"] }, Open ]], Cell[TextData[{ "So our variable is the key k, with 9 variables. 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and hardware systems, S-boxes \ should not be very large, otherwise they require a large amount of memory to \ store. 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In \ some cryptosystems (like Feistel Networks - DES), S-Boxes do not need to be \ bijective. \ \>", "Text", CellChangeTimes->{{3.728544965470003*^9, 3.728545042117416*^9}, { 3.728545096357563*^9, 3.728545105370483*^9}, {3.728548316420665*^9, 3.728548381060697*^9}},ExpressionUUID->"a365172b-a5d0-49f1-b190-\ 44b884f7839c"] }, Closed]], Cell[CellGroupData[{ Cell["Substitution-permutation network", "Subsubsection", CellChangeTimes->{{3.728545059798962*^9, 3.7285450674529448`*^9}, { 3.728545157612381*^9, 3.728545157929068*^9}, {3.728545537794989*^9, 3.7285455919343567`*^9}},ExpressionUUID->"a34b8ce9-46e3-44f2-923c-\ 5b3bbff005ec"], Cell[TextData[{ "Some modern cryptosystems (like AES) are based on substitution permutation \ networks:The idea is the following:\n\n1) Split the encryption in rounds \ i=0... r-1. \n\n2) Consider a function called the key scheduler (ks), that \ for each round i and given the key k generates a key round.\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"ks", RowBox[{"(", RowBox[{"k", ",", "i"}], ")"}]}], "\[RightTeeArrow]", SubscriptBox["k", "i"]}], TraditionalForm]],ExpressionUUID-> "ced6b3ff-0024-43bd-8662-d17efe4fc737"], "\nand so we will get key rounds ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["k", "0"], ",", " ", RowBox[{ RowBox[{ SubscriptBox["k", "1"], ".."}], RowBox[{ SubscriptBox["k", RowBox[{"r", "-", "1"}]], "."}]}]}], TraditionalForm]],ExpressionUUID-> "8b41126c-1064-4d41-8eac-2958bc572327"], "\[LineSeparator]\n3) Encryption of the message x is the composition of the \ each round encryption, that is\n\n ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"enc", RowBox[{"(", RowBox[{"k", ",", "m"}], ")"}]}], "=", RowBox[{"renc", "(", RowBox[{ SubscriptBox["k", RowBox[{"r", "-", "1"}]], ",", RowBox[{"renc", "(", RowBox[{ SubscriptBox["k", RowBox[{"r", "-", "2"}]], ",", RowBox[{ RowBox[{"...", " ", RowBox[{"renc", "(", RowBox[{ SubscriptBox["k", "0"], ",", "x"}], ")"}]}], "..."}]}], ")"}]}], ")"}]}], TraditionalForm]],ExpressionUUID-> "c02038cc-1138-4510-a31f-a4e49e2fde2b"], ".\n \n 4) Each round encryption is the decomposed in 3 parts:\n \n a) \ The key round is summed (modulo 2) to the message\n b) S-boxes are applied \ to the result: \n \t- usually the text is split into several blocks and an S \ box is\n \t applied to each block.\n c) A P permutation is applied to the \ result (which is a linear transformation). \n \n that is ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"renc", "(", RowBox[{ SubscriptBox["k", "i"], ",", "y"}], ")"}], "=", RowBox[{"P", " ", ".", " ", RowBox[{"Sf", "(", " ", RowBox[{ SubscriptBox["k", "i"], "\[CirclePlus]", "y"}], " ", ")"}]}]}], TraditionalForm]],ExpressionUUID->"a23df608-7f14-4bde-b7f0-7d4df132c7a6"], ".\n \n 5) Sometimes the last round (or some rounds) might be different \ from others in order the simplify the decryption method.\n \n An \ illustration of substitution permutation network is given below (extracted \ form Wikipedia)." }], "Text", 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