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RSA-OAEP

karapto profile image Shitian Daxiang ・5 min read

Existing RSA ciphers do not satisfy indistinguishability under adaptively chosen ciphertext attacks.
The RSA-OAEP cryptosystem is an adaptation of the RSA cryptosystem that is secure against adaptively chosen ciphertext attacks.

The security of public key cryptography is determined by two models: the decryption model and the attack model.

The attack model is a specific model that considers the assumptions of whether the attack method is unknown or known, plaintext or ciphertext, and can be roughly divided into Ciphertext-Only Attack, Known-Plaintext Attack, Chosen-Plaintext Attack, Adaptive Chosen-Plaintext Attack, Chosen-Ciphertext Attack, and Adaptive Chosen-Ciphertext Attack.

  1. Ciphertext-Only Attack(COA)
  2. Known-Plaintext Attack(KPA)
  3. Chosen-Plaintext Attack(CPA)
  4. Adaptive Chosen-Plaintext Attack(CPA2)
  5. Chosen-Ciphertext Attack(CCA1)
  6. Adaptive Chosen-Ciphertext Attack(CCA2)

On the other hand, the decoding model is a property that guarantees the difficulty of decoding a ciphertext, and is divided into Onewayness (OW), Semantic Security (SS), Indistinguishability (IND), and Non-Malleability (NM). However, SS and IND are known to be equivalent.

  1. Onewayness
  2. Semantic Security
  3. Indistinguishability
  4. Non-Malleability

The first step is to load the Crypto module. If you don't have it, or if you want to run it in Google Colab or other environment, you need to install it by pip.

!pip install Crypto 
!pip install pycrypto 

import Crypto.PublicKey.RSA as RSA
from Crypto.Cipher import PKCS1_OAEP
from Crypto.PublicKey import RSA
from Crypto.Hash import SHA512
import binascii
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After the module is loaded, the next step is to create an instance to implement the RSA cipher, setting the bit to 9216 and setting the e, d, p, q, n, and u values to be used in the RSA cipher.

rsa_fact = RSA.RSAImplementation()
rsa_key = rsa_fact.generate(bits=9216) 
print(f"e = {rsa_key.e}")
print(f"d = {rsa_key.d}")
print(f"p = {rsa_key.p}")
print(f"q = {rsa_key.q}")
print(f"n = {rsa_key.n}")
print(f"u = {rsa_key.u}")
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e = 65537
d = 54712778709843758446774472112947462605632784372693122692337276261706132343656126613850926869326406853031970937071342771082822051125805156814265100039334033193394094696992133614788847524124626876788542754592152526755990085909097407899383965497160111603490967247421144341240388433715972885769118066232454391706407281337508657486450869604612564915186037245926381485539948826749394883310642506522860818664271873435204459633736609361644611422172966074295803766199657927782114843907265391913377990158502268099171524921346814639644811634440213721098581488684823081553716072399909286975689372830754895978161079557740719349489560157014980070304726939194392515589389021598696807451490025331025472197541696176574536959564303309782584754250421256847013629445055202895707583429013726974641572883140706464114649362282002536217020801728046128596891191308549622359954181692893371397911637301634686429007441755812855730453915884304296900810588878381229626826762914932745596513434539904181264409311386739871120866568366112158792401213076791241640316240227716212342340269517372470349100996450241482980418865133120619705113594803125536458461903619411425887272770066832982698961431770946272179026490471026816477218733565162061250660114444677999978434132079969423273815362068487751387394001307409566746300482655950466254595004445848729603426974921743117321067583048597487994378747311788421673087176130320191643106706361050698733121878677309152032982200158816636173380076359282229190818958256050449423833086070508663202237289102343430444821667542376265505316574669575266888317615128990541614257673495565638717070299695356619528982437194031623223088997989273538527254541439966436390623411911288976594498254117975827186468761820788181917166160655908222038793192188787506057536765607981183236859861486667943287248863826085430979006295460903969125791512868790008624781444689451536552698147817497993529092563912042411447572632795718879409128809355718412607317831057712125337650191691644798974837098670800670701720810017663061318642521588450469140323451770630715595160815378232359863222296870117263678477582812648780595241664724700622530881004222017746966822917700708971417727097797617274650711675753090288153743294368349630966560446422220442000918908188890792339274543542052469683247718010214669699817444469516823417411264300171803938680624000616913222168448190238169463586309217889800457740939560491028217682579223583692099585650500567564310601363376397288638834272224138612305281785850918201163739271142648677843587525732067833869534084619217243476545179795365399703843687277297783992127210539155348404705206088773242790273376856944095440344175614153608432945927894754645537955829484068839908334168255572223279897229878546502328722404499946139267249915380158535466834444509333020445069897893353705865
p = 12130072046886278833181693082100614705209490447837726916478795788322960425187433549273910811163858383006895211566493794758543314214469285471236812510994877816735418918530066077217991730693600041466398450508367310195026540236275364218960767726620386759305379726879156874372965815470952269630302500173347327737693854988729268125509953762486822230357042090218143277836388276304773217375314690742789099130276227817495338520710275364341485155649268557827356048679509841643856093386007904154178133172759581155257670650028891335450188812289959428036381574569572848288535147607757369434016439509738989561427640240846187462354047641648721167182151585825057563180217175344933304090734367449738582350990303309378180110441897589353963213209371167770839804116617001521473687963876715344669709618814276909295668483541284565699994873849455930797486697856953941240424898746290877655182663159735474084712028149263972582942278156444963327822302574036936538681387111065077622844779223675297781440327259496102826382911113441313895030709814775627366783374611349731038590334887254584403840920672527132497102789239488068413183210713915507467132039515076945406086964071206518087086227370462515386332255612649671257472939512949302542353425786572795169262367563943848996139195471409494525329479762489213230006644732987840867198923512825208227632540582309134086997973750198226191910215295385938719089111438470634629
q = 12519805089236855940644210164209487623056371482442480033891581785951665924652122387636234172969095523357832081173146423080702422310354655826250129936269506566322422748900965223409189685589013330288514205723610286815010104443004435906187760806971871410953314908142729011749321942800951734468583784583950464476258455128461462235794149719745985882483978225051023582170954741174070891339318418665713855921985629953661579695382235034076281235217675973909551120871901574560206528446274641484268876807337164011673574772349829718276881142444789314205845707277490045469191728542280765995963818552845065797299975252921830778632300367309810556300770135093908113204393740605060086069333971785454040389147500355972790011429678646857816729014936550909634890407619686980073019768893724302757522559371788273933025350731188260796680491797869578748755298530752042259929172172073220561245954937796837582346075129004838903435323759761168335338503154239996870811663325163467144633044444683305828032595987699706984427334806391388798223770770759146765627296261487267519617152876345941957791634282765073377692982823376414652977870529648753561689121940883129724099795198962660636134643368278840349884630792737349208594102964824563232263117048499819487289804609056850410444299774680189540308196827320365488901128335582300667298664516455480259598865327627717566900947312780072998889331304668363400401827709600071239
n = 151866137745416559964688432462252249239140899980229095840401002683640455525229408745582490967559388671685073876703425995868659047250514275640019137744180023438036033380914550875118237270278923875485609525547664230486100642083118750645967004649840423283977278408125006848158542068546174029674757134669279720014519249375981740955043439129113280540703372283523665385533506681575328976643495741391246854127329878841344910042615567647880348090930061310877306824167844716998706557331347761205669109356561058168455560068201524333505654994973033189599666893563900313234758808897245137458208226132276634522923326880507116344394447673130775861571339181482482838261055792152547230949485485160281918233462798793958977964464264368862871417530382360339787905338214511548684422311095363378810078439811633541090245023754843090765104920708608662481659311540740188920601296243579385934731056449842549849555745641891793062841823104046872474203700111069867691048475759431932072284145501744963259734574577644781400543487823891091049832633154617237871390683825498174930327145964170919879252551961351745800165650088065988463598732057618833716412594871261980363991170720004793830919289906082158307435479479890071215428577428318411256808772197740971775301245780566519550041818808287737185025651759082663836868185668672470752081352181931650163897914338498101726771682277579669251094827096382101189662202450247528706801595182126253780755239694355924054815116591597638863241923224957759401328356378516951298552036613701403430346061627363679308261625674587619186822019969489756671012716817953167343781500006912276898198404318225817544624942139706545510714957768405724130224044295151751258549947354541150305045790078113589355945586278683046582782171793878106819223191188021984804727473789164194402818429740290591778754298559738921522474090291994429142317273233206117290986481681738738095664842469126723615623734749386881345694943175511880029898076319845417819110458926783344321175294806398526366996857765614996690254307549487134315691991028833945558109920098447456971122417762047751568805600928997384384901221399979066351526851503503362633626878650964501615886068743543944340472537664872044563469147981087766547141773072969618866988862626405646237369965807693949174786994560543281886370835506173995971701526182793145750359730095459848775071019405310123102485875350194023236625930364934091839491472058169530377591119011866264555545140737823811539715806258841365400919854827823652643926926193766906212264774344672386155195669948088831284534824735348516353444624174191126137172153322802381084234378129779213443880989004389935095956015922766705668107643657680934715125161267797159590647680270188197204888730459846953067030083669037671381538306240840673142093309452688601121676213823304404229039280947940335331
u = 10781822158372164742305626294850686816831936241525441028001043230411582088038973667162094571894187799999749147562041473604748123629150416928301030140990157623156390638612027288675572338366206717697612690947583061508263153619828286332612888722304000580433813985435206665463379094804702738936656626553314167402691854837491124299938995328452085547321260975589522866500856536443154265563359114784492581927028546354540248880733201762730703819908738348842889637325263821921965997752379810807238918240061069181254153221962474923573838098699018734758805819683376921483153685826373880726336969800504033784742731823112064241774768842814076116633600658679876600047133526288364915626666661753097150236067251142522143737634832651262466849731567407483241432201831247209145398769142292010779623659758305502902229581712596101308986885819673232808246175699681672202309156701075927859899886829767012503005194934421294783981572466821103049254421135653208178444947763051815002282479680402390074943146793557595444822691075774625011960189188631315907051066786786183550298038765045896809870184328342874576998223471703093795866581191011541245524753621535560250397713625184569644431536651498541155015903163493369214406453958554598704502289203541177059659769715495474329271405160176325178717639725228119763163665086490064099329801540922235306326637478727213854429773502229254532779759856101596122507840306436792107
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Next, each of the values created above is encrypted. The hash function used for the encryption is SHA512.

message = u'This is plain Text'.encode('utf-8')
P = binascii.b2a_hex(message)
key = rsa_key
cipher = PKCS1_OAEP.new(key, hashAlgo=SHA512)
ciphertext = cipher.encrypt(P)
ciphertext
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b'G\x82\xed\xe7\x00rH\xb0KF\x8c\xd7-\xafz@\xe0\x8b\x01\x7f\x1f\x8f\x19\x12- \x83\xc6b\xd6V)^\x10\x14\xaf\xa11L=\x93\xba\xa9\x86r\xda2\x99\xa1q\xc8\x97X\xf7P\x8d\x94\x19\xf9\x13*\x13c\x8fD\xa3\x90\x18\xcc\xf9\xcf\xa1jF\x91Z\xe3^r\xb8\x08\xb2;\xb4\x8f\xd0\tf\x18\xee\xda\x0c\x81\xa1S\xf1\xbaZ\x00\xdf\xab\xcf\xb2\xbeW\xc4\xb1\xba\x8a\xbb\xd6\x96\x1f\xac<Q\xa93<\xaf\x99\xc1\x82\x86\xe0\xb3\xbf\xb6,\xf0\xf7\xe0\x98\x19\xa2:\x1fC\xa5W_\r\x99P\xf9\x80\xe3\xeeXy\x14\x88\xfaZN\xd3\xa3!\xcdv\x98\x9fi\xc1\xb3\xd06\xefKx7\x1c\x8d\xd8\xa5z\xb1|\xf8k\xdd\xfc\xce!F\xcb\xfd\xff++\xa7\xf9\xed.\\\x14\xa9\xadc\xd7\xbf@sW\xa5\x94\x9a\x8eB\x05\nk\xc8,\x19\xd9\xb6\x8aY\x92\x86\x17Y\x89\x02\xbfJ\xfe\xd9\xf8A=@\xba\x87\xc2t\xdb\x1f\xd3K\x0f+3\xdfK-4B\xca\xfe}\xd6o\xef\x04GL!\x9f\xee\xe0\xd4{\xee\x07F\xf1\xc9\x82F\xe2Y\xa5\x15\x11.\x88"\x12c;\x84m(\x15h\x89\xba\x1b\xab\x8a\xdc\x844\xfd+\xaa\xbe\r/b\xa5\x08\xa9\xa4\xbb\xb9\xe5\xcdl\xcc_\x9f\x01\x9dP\xed\xe7\x9db\xffF\x14+y\x07\x95\xc8\xd4c\xe7\x98SZC\xfc\xc30\xcf\xa7\x1a6bAUqp\xeehP\xa7Hr\x80UP\xd7\x8dM\x97\xd9\xf0\x18\x02t"\xa0\xeawdL\xc1>"\x1e\xed!\xfc+\x1f\xa3\x98\x1acXd;\xb0\xd4\x904"\x9a\xf6l\xc6\t\x8a\x9d\xb3\x83\xb0p\xc0\x8f6\xe3\xcb\x185\xf3\xba\xf1ux\xe3\x81\x05\xad\xe4>{\xf4w\xdaN\x85\n\x87 \xf6b,Dk\xb0P\xc7\x9e4\xca\xed[9\x14\x89tRF`\xcfC\xc52\xf5\x022\x94! \xef\xd6lU\x8ev\xec(V\xff\xe0M\'vM5\x8e\x8d\x96\xd0Sc~g\xb5\x03a\x1e\xfd_q\xea\xf0\x8b\x9e+sTE\x14\xfd\x8fr\x16a\\\x1cC\xeev\x0e\x13nv\x9e\xd4\x94,\xbc\xa5F6\x9d\xfa`\x04\xf3Xl\x03\x9f\xad\xb8b\x9fI\xf1\xad-\xb1\x06\xbb\x9a\x13C\xa4^\x89O\xf4S\xf5M\xe2m!\x94\xc2\xee<U\xa1xo\x9c\x01r\x8f\xce\\\x11\x18\n\x0cN\x9bhyQ\x03\x04\x005z3\x0b^\xe0\xcc&\xa5\x01\x0c!\x10\x9a\xed5\x87$,! \xe8vq*\x17\xa2\xe4\x05\x93\x015\xa7%\x89\xc6\x0f\x80\xb8\xb1e`I\xc5^\xb7?\xba\xdfh\x1a\xdd+\x14\xd2\x13\xc4\xf1\x98\xfa\xde\xc8#A2\xad\xcf\xcd\x7f\x85\xd1\x8f\x12Z:\x1b\xe6\xcf\xe4-I\xe0M\x93\x0f\x93\xcemq\xedP\xd0zX\x8b7\x84\xc7\n\x96\x95\xbd\xa5D\xaf\xca\x8e_\x9cw\x05\x1d\xb8~4aE\x0c\x96\xe3\xc8"A\xfb"\xc0\xfb\xb2\xf3\xb7\x06\xea\xaa\n\xf17\xe0]uUk\xdd\x85N\xa8\xbc\xeaF\xe8\xc6u\xcb;\x89I_q\xe0^9%\x9ae\x82\x849=\xb4)\x04\xd9\xb9*\xdf\xe1\x93\x84\x83\xef0\xb6\xa5\xf0\xdd<\xce?\xadh\xc6\xb6\x85\xc7\xa1\x9b&\xad\xf5\xb3\xca\x19\x9d\xccT\x11r\xe5\xc0\xc6\x0b\xb8,:~\xe7\xcc\x12u\x8a\xa6\xe0\xce\xf5\xca \xfd\x00\x88\xf5\xf93\xb4\xe8\x89\xd94\xf7\xbf`\xc3\xa5\xc7\x9e\xd3Y\x1a\xda\xcfC\xad\x80l%b\xa0\xd8R2\x88\xfeM\x08\xdc?\xc1\xa2\x01F\xa9\\8\x07i\xc0\x04\xe3\x1f\xd3\xa5\xfa\xde\\[\xd3\x1eg2I\xa4\xcd{Ej\xff\x14\xf7g\xac\xc0\xf7\x82^=\xad\x88\x83\xb8\xa8\x84\xe1\x98\xf8\xc8\xa9\xe2TV\x82\x1cfy\xf5\xcf)oB\x18\x85<\xb7\xe9\x8a\x84\x90\xb8\xd6;\xba\xf0\xc6\xecf\x8b\xdf_,9\xf2\xbf\xd6IuI\xf2\xb8P\x06I\xa5\xaa\xd5WY\xbd\xd1\xf6\xf1B\xba\x11\xd5\xf7\xc7\xd2%\xfc\xa8.\xe0\x15\xef\xbf\xdb\xd1\xebNp \x86\xaa\x15\x1dv\xee!C,\xa1$\xa5\xe9_aI\x01\xb8\xb0\xb9\xa6Z\x97\xf1l\'\xee\x82\x04\xcb\xaf\\p\x0ef\x9b\xb8 \xdcoD\xb8\xc6\xa3\x1c\xe5\x17[\xb4\xc7Q}(\xb8\xb9G\x81\xd6jj\xea(\xb5\xf2\xe6\x84\xd0n\xf9}\x1e\x9f\xe4\xbc\xae\x1d\xd1\xe0\xbf\x97\xfa^\xcb\xb9oc/\x84H$\xe2\xb33\xa6\xd9\xc6I\xe7\xbe\x01[\xff\x95\x12\xcf\xc8rl\xb4\x14\x80>f|\xf0\xb5E\xe4==<b/\xbe\x90\xaa\xbfp\x80\x84\xa2\xa8\xe2o\xf6lR\xfaQ\xa0w\xd2\xbd\xecU<Z\xa2\xc1\xbf\x8b\xff\x1eN\x88[\x9c\xa8\xe8i\x7f\xea\x84\x80\xd6H\x91\xa5\xa6\xbe\xff\x18\x0197\xe239\x16\x0f\x8f'
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Finally, we decrypt the ciphertext we created above. When running the program in a notebook, you can measure the execution time by adding the magic command %%time to the top of the cell, and you will see that it takes almost 10 times longer than the encryption case.

M = cipher.decrypt(ciphertext)
message = binascii.a2b_hex(M).decode('utf-8')
message
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This is plain Text
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