October 28, 2004Secure Information Transmission (network security model)Security of Information Systems (Network access model)Issues in Network securityCryptologyElementary Number TheoryDivisorsPrime NumbersGreatest common divisor (GCD)Relatively Prime NumbersThe modulo operationModular ArithmeticModulo OperationBrief HistoryCryptosystemExampleCæsar cipherAttacksAttacking a conventional cryptosystemBasis for CyptoanalysisClassical CryptographySlide 22Slide 23Transposition CipherAttacking the CipherSlide 26Slide 27Substitution CiphersSlide 29Statistical AttackCharacter Frequencies (Denning)Statistical AnalysisCorrelation: (i) for 0 ≤ i ≤ 25The ResultCæsar’s ProblemVigenère CipherRelevant Parts of TableauUseful TermsAttacking the CipherOne-Time PadOverview of the DESDESEnciphermentThe f FunctionControversyUndesirable PropertiesDES ModesCBC Mode DecryptionSelf-Healing PropertyCurrent Status of DESPublic Key CryptographyRequirementsDiffie-HellmanAlgorithmSlide 55RSASlide 57Confidentiality using RSAExample: ConfidentialitySlide 60Authentication using RSAExample: Origin Integrity/AuthenticationSlide 63Confidentiality + AuthenticationExample: Confidentiality + AuthenticationSlide 66Security ServicesMore Security ServicesWarningsCryptographic ChecksumsMathematical characteristicsDefinitionCollisionsKeysMessage DigestHash Message Authentication Code (HMAC)Security LevelsAverage time required for exhaustive key searchKey PointsIN2935/TEL2810: Introduction of Computer Security1October 28, 2004October 28, 2004Introduction to Introduction to Computer SecurityComputer SecurityLecture 7Lecture 7Basic Cryptography & Network SecurityBasic Cryptography & Network SecurityIN2935/TEL2810: Introduction to Computer Security 2Secure Information TransmissionSecure Information Transmission(network security model)(network security model)Trusted Third Partyarbiter, distributer of secret informationOpponentSecure MessageSecure Message MessageInformation channelSender ReceiverSecret InformationSecurity related transformationSecret Information MessageIN2935/TEL2810: Introduction to Computer Security 3Security of Information SystemsSecurity of Information Systems(Network access model)(Network access model)GateKeeperOpponent - hackers - softwareAccess ChannelInternalSecurity ControlDataSoftwareGatekeeper – firewall or equivalent, password-based loginInternal Security Control – Access control, Logs, audits, virus scans etc.IN2935/TEL2810: Introduction to Computer Security 4Issues in Network securityIssues in Network securityDistribution of secret information to enable Distribution of secret information to enable secure exchange of information is importantsecure exchange of information is importantEffect of communication protocols needs to be Effect of communication protocols needs to be consideredconsideredEncryption (cryptography) Encryption (cryptography) if used cleverly and if used cleverly and correctlycorrectly, can provide several of the security , can provide several of the security services services Physical and logical placement of security Physical and logical placement of security mechanismsmechanismsCountermeasures need to be considered Countermeasures need to be consideredIN2935/TEL2810: Introduction to Computer Security 5CryptologyCryptologyCRYPTOLOGYCRYPTOGRAPHY CRYPTANALYSISPrivate Key(Secret Key)Public KeyBlock Cipher Stream Cipher Integer FactorizationDiscrete LogarithmEncipher, encryptDecipher, decryptIN2935/TEL2810: Introduction to Computer Security 6Elementary Number TheoryElementary Number TheoryNatural numbers N = {1,2,3,…}Natural numbers N = {1,2,3,…}Whole numbers W = {0,1,2,3, …}Whole numbers W = {0,1,2,3, …}Integers Z = {…,-2,-1,0,1,2,3, …}Integers Z = {…,-2,-1,0,1,2,3, …}DivisorsDivisorsA number b is said to divide a if a = mb for some m where a,b,m  ZWe write this as b | aRead as “b divides a”IN2935/TEL2810: Introduction to Computer Security 7DivisorsDivisorsSome common propertiesSome common propertiesIf a | 1, a = +1 or –1If a|b and b|a then a = +b or –bAny b  Z divides 0 if b  0If b|g and b|h then b|(mg + nh) where b,m,n,g,h  ZExamples: Examples: The positive divisors of 42 are 1,2,3,6,7,14,21,423|6 and 3|21 => 3|21m+6n for m,n  ZIN2935/TEL2810: Introduction to Computer Security 8Prime NumbersPrime NumbersAn integer An integer pp is said to be a prime number if its only is said to be a prime number if its only positive divisors are 1 and itselfpositive divisors are 1 and itself1, 3, 7, 11, ..Any integer can be expressed as a Any integer can be expressed as a uniqueunique product of product of prime numbers raised to positive integral powersprime numbers raised to positive integral powersExamplesExamples7569 = 3 x 3 x 29 x 29 = 32 x 2925886 = 2 x 27 x 109 = 2 x 33 x 1094900 = 72 x 52 x 22100 = ?250 = ?This process is called This process is called Prime FactorizationPrime FactorizationIN2935/TEL2810: Introduction to Computer Security 9Greatest common divisor (GCD)Greatest common divisor (GCD)Definition: Greatest Common DivisorDefinition: Greatest Common DivisorThis is the largest divisor of both a and bGiven two integers Given two integers aa and and bb, the positive , the positive integer integer cc is called their GCD or greatest is called their GCD or greatest common divisor if and only ifcommon divisor if and only ifc | a and c | b Any divisor of both a and b also divides cNotation: gcd(Notation: gcd(aa, , bb) = ) = ccExample: gcd(49,63) = ?Example: gcd(49,63) = ?IN2935/TEL2810: Introduction to Computer Security 10Relatively Prime NumbersRelatively Prime NumbersTwo numbers are said to be relatively prime if Two numbers are said to be relatively prime if their gcd is 1their gcd is 1Example: 63 and 22 are relatively primeHow do you determine if two numbers are How do you determine if two numbers are relatively prime?relatively prime?Find their GCD orFind their prime factorsIf they do not have a common prime factor other than 1, they are relatively primeExample: 63 = 9 x 7 = 32 x 7 and 22 = 11 x 2IN2935/TEL2810: Introduction to Computer Security 11The modulo operationThe modulo operationWhat is 27 mod 5?What is 27 mod 5?DefinitionDefinitionLet a, r, m be integers and let m > 0We write a  r mod m if m divides r – a (or a – r) and 0  r < mm is called the