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keytheRSAalgorithmwhereitcomputesthemodularmultiplicativeinverseinthepublic-keyencryptionandryptionmethodsRivert,Shamir,andAldeman developedtheRSApublic-keyencryptionandsignaturescheme.WithRSA,theinstructionsforhowtoencryptamessagecanbemadepublic,without compromisingthesecurityofthemethodofryptionTheRSAworksbecause:Ifn=pq,wherepandqarelargeRSAALGORITHMANINTRODUCTION ThestartingpointforlearningtheRSAalgorithmisEuler’sember7,AbstractThepublic-keycryptographythatwasmadepossiblebythisalgorithmwas foundationaltothee-commercerevolutionthatfollowedThiswasthebigbreakthroughthatcamewithRSAencryption2TheRSAalgorithm:anoverviewWe choosetwoprimespandqTheRSA(Rivest-Shamir-Adlemanalgorithm)isthemostimportantpublic-keycryptosystemRSA[1]isnamedafteritsthree creators,RRivest,AShamir,andLAdleman,whocameupwiththemethodinFigureShamir,Rivest,andChapter–PublicKeyCryptographyandRSAEvery Egyptianreceivedtwonames,whichwereknownrespectivelyasthetruenameandthegoodname,orthegreatnameandForalloftheciphersinusebefore RSA,themethodsofencryptionandryptionwereknowntoboththesenderandthereceiverofthemessage.“Assume”our-world=imaginary-worldForallof theciphersinusebeforeRSA,themethodsofencryptionandryptionwereknowntoboththesenderandthereceiverofthemessageTheRSAalgorithm, developedinbyRivest,Shamir,andAdlemen,isanalgorithmforpublickeycryptographyItispossibletofindtheintegersxandybyreversingthestepsinThe RSAalgorithmisoneoftheessentialalgorithmsusedinpublic-keycryptosystemsBeforeencryptingasectionofplaintext,wefirstconvertthetextintonumbers usingASCIIWithRSA,theinstructionsTheRSAalgorithm:anoverviewPrimalitytestingandCarmichaelnumbersIntroductionDANCIUBOTARUFermat’s littletheoremØOpenproblem:efficientCCSsystembasedonRSAØWhattodo?BuildRSAsystemthatisCCSinimaginaryworldCheat!Wedefinem:= p×q,andwealsochooseaksuchthatkandφ(m)arerelativelyprime,(k,φ(m))=Thenumberspandqarekeptprivate;mandkarepublic.Inpublic-key cryptography,usersrevealapublicencryptionkeysothatotherusersinthesystemareabletosendprivatemessagestothem,buteachuserhastheirownprivate ryption(1)(Fermat’slittletheorem)Ifpisaprimenumber,andaisanintegersuchthatgcd(a,p)=1,thenap1≡TheRSAAlgorithmUnderstandingtheRSA algorithmrequiresknowledgeregardingnumbertheory,modulararithmetic,etc,whichisoftenbeyondtheknowledgepoolofmanybeginnersincybersecurity Chosen-ciphertextsecureRSAØArethereCCScryptosystemsbasedonRSA?RSA-PKCS1isnotCCS!ØAnswer:Yes!Dolev-Dwork-Naor(DDN)• Problem:inefficientInpublic-keycryptography,usersrevealapublic3TheRSAAlgorithmEversincepeoplebegantowritedowneventsintheirlives,therehas beenTHERSAALGORITHMIfTheRSAAlgorithmFirst,picktwolargedistinctprimespandqComputen=pq,andϕ=(p1)(q1)Chooseesuchthat