fair wear foundation dedicated sustainability fairness garment industry mentioned initiatives sustainability encompass green topics well social sustainability however many different ways sustainability initiatives implemented company improving operations organization implement sustainability initiatives improving operations manufacturing process make aligned environment social governance issues johnson johnson incorporates policies universal declaration human rights international covenant civil political rights international covenant economic social cultural rights applying


principles members supply chain also internal operations walmart made commitments doubling truck fleet efficiency replacing rds fleet fuel efficient trucks including hybrids dell integrated alternative recycled recyclable materials products packaging design improving energy efficiency design end life recyclability dell plans reduce energy intensity product portfolio ref name theguardiancom ref board leadership board company decide lower executive compensation given percentage give percentage compensation

Applied science
Business
Business ethics


specific cause effort implemented top affect compensation executives company alcoa aluminum company based us th executive cash compensation tied safety diversity environmental stewardship includes greenhouse gas emission reductions energy efficiency best practices usually case companies see board take uniform step towards environment social governance issues case companies directly linked utilities energy


material industries something alcoa aluminum company falls line instead formal committees focused environment social governance issues usually seen governance committees audit committees rather board directors according research analysis done pearl meyer support nacd director compensation report shows among public companies reviewed slightly five percent boards designated committee address esg issues compensation ref name theguardiancom management accountability similar board leadership creating steering committees types committees specialized


sustainability senior executives identified held accountable meeting constantly improving sustainability goals ref name theguardiancom executive compensation introducing bonus schemes reward executives meeting non financial performance goals including safety targets greenhouse gas emissions reduction targets goals engaging stakeholders help shape companies public policy positions companies exelon implemented policies like ref name theguardiancom stakeholder engagement companies keep sustainability within strategy goals presenting findings shareholder meetings actively tracking metrics sustainability companies pepsico heineken fifco take steps direction implement sustainability initiatives


best practices companies coca cola actively tried improve efficiency water usage hiring rd party auditors evaluate water management approach fifco also led successfully led water management initiatives ref name theguardiancom employee engagement implementation sustainability projects directly appealing employees typically human resource department another option companies implement sustainability involves integrating sustainability company culture hiring practices employee training general electric company taking lead implementing initiatives manner bank america directly engaged employees implement leed leadership


energy environmental design certified buildings fifth building meeting certifications ref name theguardiancom supply chain management establishing requirements internal operations also first tier suppliers well second tier suppliers help drive environmental social expectations supply chain companies starbucks fifco ford motor company implemented requirements suppliers must meet win business starbucks led efforts engaging suppliers local communities operate accelerate investment sustainable farming starbucks set goal ethically sourcing coffee beans ref


journals appeared researchers entered field corporate scandals earlier increased field 's popularity sixteen academic journals devoted various business ethics issues existed ''journal business ethics'' ''business ethics quarterly'' considered leaders ''journal business ethics education'' publishes articles specifically education business ethics international business development institute global non profit organization represents nations united states offers charter business development focuses ethical business practices standards charter directed harvard mit fulbright scholars


includes graduate level coursework economics politics marketing management technology legal aspects business development pertains business ethics ibdi also oversees international business development institute asia provides individuals living asian nations opportunity earn charter religious views sharia law followed many muslims banking specifically prohibits charging interest loans traditional confucian thought discourages profit seeking christianity offers golden rule command therefore things whatsoever ye would men ye even law prophets


according article theory real economy narrow point view christianity faith towards relationship ethics religious traditions article stresses christianity capable establishing reliable boundaries financial institutions one criticism comes pope benedict describing damaging effects real economy badly managed largely speculative financial dealing mentioned christianity potential transform nature finance investment theologians ethicist provide evidence real economic life business


ethics receives extensive treatment jewish thought rabbinic literature ethical ''mussar'' legal ''halakha'' perspective see article ''jewish business ethics'' discussion according article indian philosophy business ethics review chandrani chattopadyay hindus follow dharma business ethics unethical business practices termed adharma businessmen supposed maintain steady mindedness self purification non violence concentration clarity control senses books like bhagavat gita arthashastra contribute lot towards conduct ethical business related disciplines business ethics related philosophy economics branch philosophy deals


requirements social justice natural rights especially property rights relation business enterprise business ethics also related political economy economic analysis political historical perspectives political economy deals distributive consequences economic actions see also b corporation certification business culture business law corporate behaviour corporate crime ethical code ethical consumerism ethical implications contracts ethical job ethicism evil corporation moral psychology optimism bias organizational ethics penny stock scam philosophy economics political corruption strategic misrepresentation strategic planning notes references display authors etal reading applied ethics industrial organizational psychology


telemedicine system ''federal center neurosurgery tyumen '' hemodialysis process purifying blood person whose kidneys working normally '''biomedical engineering''' '''bme''' '''medical engineering''' application engineering principles design concepts medicine biology healthcare purposes eg diagnostic therapeutic bme also traditionally known bioengineering term come also refer biological engineering field seeks close gap engineering medicine combining design problem solving skills engineering medical biological sciences advance health care treatment including diagnosis

Applied science
Engineering and technology
Mechanical Engineering
Biomedical engineering


monitoring therapy ref name enderlebronzino ref also included scope biomedical engineer management current medical equipment hospitals adhering relevant industry standards involves making equipment recommendations procurement routine testing preventive maintenance role also known biomedical equipment technician bmet clinical engineering biomedical engineering recently emerged study compared many engineering fields evolution common new field transition interdisciplinary specialization among already established fields considered field much work biomedical


engineering consists research development spanning broad array subfields see prominent biomedical engineering applications include development biocompatible prostheses various diagnostic therapeutic medical devices ranging clinical equipment micro implants common imaging equipment mris ekg ecgs regenerative tissue growth pharmaceutical drugs therapeutic biologicals bioinformatics example approximately probe spotted oligo microarray enlarged inset show detail '''bioinformatics''' interdisciplinary field develops methods software tools understanding biological data interdisciplinary field science bioinformatics combines computer science statistics mathematics engineering analyze


interpret biological data bioinformatics considered umbrella term body biological studies use computer programming part methodology well reference specific analysis pipelines repeatedly used particularly field genomics common uses bioinformatics include identification candidate genes nucleotides snps often identification made aim better understanding genetic basis disease unique adaptations desirable properties esp agricultural species differences populations less formal way bioinformatics also tries understand organizational principles within nucleic


acid protein sequences biomechanics ribosome biological machine utilizes protein dynamics biomechanics study structure function mechanical aspects biological systems level whole organisms organs cells cell organelles using methods mechanics biomaterial '''biomaterial''' matter surface construct interacts living systems science '''biomaterials''' fifty years old study biomaterials called '''biomaterials science biomaterials engineering''' experienced steady strong growth history many companies investing large amounts money


development new products biomaterials science encompasses elements medicine biology chemistry tissue engineering materials science biomedical optics biomedical optics refers interaction biological tissue light exploited sensing imaging treatment tissue engineering tissue engineering like genetic engineering see major segment biotechnology overlaps significantly bme one goals tissue engineering create artificial organs via biological material patients need organ transplants biomedical engineers currently researching methods creating organs researchers grown solid jawbones ref


name bbc ref tracheas human stem cells towards end several artificial urinary bladders grown laboratories transplanted successfully human patients ref name cnngrow ref bioartificial organs use synthetic biological component also focus area research hepatic assist devices use liver cells within artificial bioreactor construct ref name chicagoliver http wwwuchospitalsedu news eladhtml trial begins first artificial liver device using human cells university chicago february ref micromass cultures c h cells varied oxygen tensions stained alcian blue genetic engineering genetic engineering recombinant dna technology genetic modification manipulation gm


gene splicing terms apply direct manipulation organism 's genes unlike traditional breeding indirect method genetic manipulation genetic engineering utilizes modern tools molecular cloning transformation directly alter structure characteristics target genes genetic engineering techniques found success numerous applications examples include improvement crop technology ''not medical application'' see biological systems engineering manufacture synthetic human insulin use modified bacteria manufacture erythropoietin hamster ovary cells production new types experimental mice


oncomouse cancer mouse research neural engineering neural engineering also known neuroengineering discipline uses engineering techniques understand repair replace enhance neural systems neural engineers uniquely qualified solve design problems interface living neural tissue non living constructs pharmaceutical engineering pharmaceutical engineering interdisciplinary science includes drug engineering novel drug delivery targeting pharmaceutical technology unit operations chemical engineering pharmaceutical analysis may deemed part pharmacy due focus use technology chemical agents providing better


medicinal treatment medical devices schematic silicone membrane oxygenator ''extremely broad category'' essentially covering health care products achieve intended results predominantly chemical eg pharmaceuticals biological eg vaccines means involve metabolism medical device intended use diagnosis disease conditions cure mitigation treatment prevention disease examples include pacemakers infusion pumps heart lung machine dialysis machines artificial organs implants artificial limbs corrective lenses cochlear implants ocular prosthetics facial prosthetics somato prosthetics dental implants biomedical instrumentation amplifier schematic used


monitoring low voltage biological signals example biomedical engineering application electronic engineering electrophysiology stereolithography practical example ''medical modeling'' used create physical objects beyond modeling organs human body emerging engineering techniques also currently used research development new devices innovative therapies treatments patient monitoring ref name curtisdalby ref complex diseases medical devices regulated classified us follows see also ''regulation'' class devices present minimal potential harm user often simpler design class ii class


iii devices devices category include tongue depressors bedpans elastic bandages examination gloves hand held surgical instruments similar types common equipment class ii devices subject special controls addition general controls class devices special controls may include special labeling requirements mandatory performance standards postmarket surveillance devices class typically non invasive include x ray machines pacs powered wheelchairs infusion pumps surgical drapes class iii devices generally require premarket approval pma premarket notification k scientific review ensure device 's safety effectiveness addition


general controls class examples include replacement heart valves hip knee joint implants silicone gel filled breast implants implanted cerebellar stimulators implantable pacemaker pulse generators endosseous intra bone implants medical imaging medical biomedical imaging major segment medical devices area deals enabling clinicians directly indirectly view things visible plain sight due size location involve utilizing ultrasound magnetism uv radiology means mri scan human head example biomedical engineering application electrical engineering diagnostic imaging click


view animated sequence slicesimaging technologies often essential medical diagnosis typically complex equipment found hospital including fluoroscopy magnetic resonance imaging mri nuclear medicine positron emission tomography pet pet ct scans projection radiography x rays ct scans tomography ultrasound optical microscopy electron microscopy implants implant kind medical device made replace act missing biological structure compared transplant indicates transplanted biomedical tissue surface implants contact body might made biomedical material


titanium silicone apatite depending functional cases implants contain electronics eg artificial pacemakers cochlear implants implants bioactive subcutaneous drug delivery devices form implantable pills drug eluting stents artificial limbs right arm example prosthesis left arm example myoelectric control prosthetic eye example biomedical engineering application mechanical engineering biocompatible materials ophthalmology bionics artificial body part replacements one many applications bionics concerned intricate thorough study


properties function human body systems bionics may applied solve engineering problems careful study different functions processes eyes ears organs paved way improved cameras television radio transmitters receivers many tools biomedical sensors recent years biomedical sensors based microwave technology gained attention different sensors manufactured specific uses diagnosing monitoring disease conditions example microwave sensors used complementary technique x ray monitor lower extremity trauma sensor


monitor dielectric properties thus notice change tissue bone muscle fat etc skin measuring different times healing process response sensor change trauma heals clinical engineering clinical engineering branch biomedical engineering dealing actual implementation medical equipment technologies hospitals clinical settings major roles clinical engineers include training supervising biomedical equipment technicians bmets selecting technological products services logistically managing implementation working governmental regulators inspections audits serving technological consultants hospital


innovation various roles form bridge primary designers end users combining perspectives close point use also trained product process engineering clinical engineering departments sometimes hire biomedical engineers also industrial systems engineers help address operations research optimization human factors cost analysis etc also see safety engineering discussion procedures used design safe systems clinical engineering department constructed manager supervisor engineer technician one engineer per eighty beds hospital ratio clinical engineers


also authorized audit pharmaceutical associated stores monitor fda recalls invasive items rehabilitation engineering ultrasound representation urinary bladder black butterfly like shape hyperplastic prostate example practical science medical science working together '''rehabilitation engineering''' systematic application engineering sciences design develop adapt test evaluate apply distribute technological solutions problems confronted individuals disabilities functional areas addressed rehabilitation engineering may include mobility communications hearing vision cognition activities associated employment independent living education integration community ref name enderlebronzino rehabilitation engineers master 's


degrees rehabilitation engineering usually subspecialty biomedical engineering rehabilitation engineers undergraduate graduate degrees biomedical engineering mechanical engineering electrical engineering portuguese university provides undergraduate degree master 's degree rehabilitation engineering accessibility ref name bbc ref name cnngrow qualification become rehab' engineer uk possible via university bsc honours degree course health design technology institute coventry university ref name chicagoliver rehabilitation process people disabilities often entails design assistive devices walking aids intended promote inclusion


users mainstream society commerce recreation schematic representation normal ecg trace showing ' 'sinus rhythm '' example widely used clinical medical equipment operates applying electronic engineering electrophysiology medical diagnosis regulatory issues regulatory issues constantly increased last decades respond many incidents caused devices patients example us fda recalls medical devices classified class according us food drug administration fda class recall associated situation


reasonable probability use exposure product cause serious adverse health consequences death regardless country specific legislation main regulatory objectives coincide worldwide ref name ref example medical device regulations product must safe ''and'' effective manufactured devices product safe patients users third parties run unacceptable risks physical hazards death injuries intended use protective measures introduced devices reduce residual risks acceptable level compared


benefit derived use product effective performs specified manufacturer intended use effectiveness achieved clinical evaluation compliance performance standards demonstrations substantial equivalence already marketed device previous features ensured manufactured items medical device requires quality system shall place relevant entities processes may impact safety effectiveness whole medical device lifecycle medical device engineering area among heavily regulated


fields engineering practicing biomedical engineers must routinely consult cooperate regulatory law attorneys experts food drug administration fda principal healthcare regulatory authority united states jurisdiction medical ' 'devices drugs biologics combination'' products paramount objectives driving policy decisions fda safety effectiveness healthcare products assured quality system place specified cfr regulation addition biomedical engineers often develop devices technologies consumer use physical therapy devices also


medical devices may also governed respects consumer product safety commission greatest hurdles tend k clearance typically class devices pre market approval typically drugs class devices european context safety effectiveness quality ensured conformity assessment defined method manufacturer demonstrates device complies requirements european medical device directive directive specifies different procedures according class device ranging simple declaration conformity annex vii class


devices ec verification annex iv production quality assurance annex v product quality assurance annex vi full quality assurance annex ii medical device directive specifies detailed procedures certification general terms procedures include tests verifications contained specific deliveries risk management file technical file quality system deliveries risk management file first deliverable conditions following design manufacturing steps risk management stage shall drive product product risks reduced acceptable level respect


accredited european member states notified bodies must ensure effectiveness certification process medical devices apart class devices declaration conformity produced manufacturer sufficient marketing product passed steps required medical device directive device entitled bear ce marking indicating device believed safe effective used intended therefore marketed within european union area different regulatory arrangements sometimes result particular technologies developed


first either us europe depending favorable form regulation nations often strive substantive harmony facilitate cross national distribution philosophical differences ''optimal extent'' regulation hindrance restrictive regulations seem appealing intuitive level critics decry tradeoff cost terms slowing access life saving developments rohs ii directive eu better known rohs recast legislation originally introduced original eu legislation restrictions certain hazardous substances electrical electronics devices rohs directive ec replaced


superseded eu published july commonly known rohs rohs seeks limit dangerous substances circulation electronics products particular toxins heavy metals subsequently released environment devices recycled scope rohs widened include products previously excluded medical devices industrial equipment addition manufacturers obliged provide conformity risk assessments test reports explain lacking first time manufacturers also importers distributors share responsibility ensure


electrical electronic equipment within scope rohs complies hazardous substances limits ce mark products iec new international standard iec home healthcare electro medical devices defining requirements devices used home healthcare environment iec must incorporated design verification wide range home use point care medical devices along applicable standards iec rd edition series mandatory date implementation en european version standard june


us fda requires use standard june health canada recently extended required date june april north american agencies require standards new device submissions eu take severe approach requiring applicable devices placed market consider home healthcare standard nzs ans australian new zealand standards management medical devices standard specifies procedures required maintain wide range medical assets clinical


setting eg hospital ref name standards australia ref standards based iec standards standard covers wide range medical equipment management elements including procurement acceptance testing maintenance electrical safety preventive maintenance testing decommissioning training certification education biomedical engineers require considerable knowledge engineering biology typically bachelor 's bsc bs beng bse master 's ms msc mse meng doctoral phd degree bme biomedical engineering another branch engineering considerable potential bme overlap interest bme increases many engineering colleges


biomedical engineering department program offerings ranging undergraduate bsc bs beng bse doctoral levels biomedical engineering recently emerging ''its discipline'' rather cross disciplinary hybrid specialization disciplines bme programs levels becoming widespread including bachelor science biomedical engineering includes enough biological science content many students use pre med major preparation medical school number biomedical engineers expected rise cause effect improvements medical technology us


increasing number undergraduate programs also becoming recognized abet accredited bioengineering biomedical engineering programs programs currently accredited abet canada australia accredited graduate programs biomedical engineering common example mcmaster university offers masc md phd phd biomedical engineering first canadian undergraduate bme program offered ryerson university four year beng program polytechnique montreal also offering bachelors 's degree biomedical engineering flinders university many degrees reputation ranking program may factor


desirability degree holder either employment graduate admission reputation many undergraduate degrees also linked institution 's graduate research programs tangible factors rating research funding volume publications citations bme specifically ranking university 's hospital medical school also significant factor perceived prestige bme department program graduate education particularly important aspect bme many engineering fields mechanical electrical engineering need graduate level training obtain entry level job


certain engineering coursework education bme also varies greatly around world virtue extensive biotechnology sector numerous major universities relatively internal barriers us progressed great deal development bme education training opportunities europe also large biotechnology sector impressive education system encountered trouble creating uniform standards european community attempts supplant national jurisdictional barriers still exist recently initiatives biomedea sprung develop bme related education professional standards countries australia


engineers us model generally require practicing engineers offering engineering services impact public welfare safety safeguarding life health property licensed engineers working private industry without direct offering engineering services public businesses education government need licensed notably case many countries license legally necessary practice engineering law medicine biomedical engineering regulated countries australia registration typically recommended


required uk mechanical engineers working areas medical engineering bioengineering biomedical engineering gain chartered engineer status institution mechanical engineers institution also runs engineering medicine health division institute physics engineering medicine ipem panel accreditation msc courses biomedical engineering chartered engineering status also sought ipem fundamentals engineering exam first general two licensure examinations us jurisdictions cover biology although technically bme


second exam called principles practices part professional engineering exam candidates may select particular engineering discipline 's content tested currently option bme meaning biomedical engineers seeking license must prepare take examination another category affect actual license since jurisdictions recognize discipline specialties anyway however biomedical engineering society bmes exploring possibility seeking implement bme specific version exam facilitate biomedical engineers pursuing licensure beyond governmental


professor emeritus university california san diego considered many founder modern biomechanics leslie geddes deceased professor emeritus purdue university electrical engineer inventor educator biomedical engineers received national medal technology president george bush years contributions spawned innovations ranging burn treatments miniature defibrillators ligament repair tiny blood pressure monitors premature infants well new method performing cardiopulmonary resuscitation cpr willem johan kolff deceased pioneer hemodialysis


published works works cited times cumulatively p hunter peckham donnell professor biomedical engineering orthopaedics case western reserve university pioneer functional electrical stimulation fes nicholas peppas chaired professor engineering university texas austin pioneer drug delivery biomaterials hydrogels nanobiotechnology robert plonsey professor emeritus duke university pioneer electrophysiology otto schmitt deceased biophysicist significant contributions bme working biomimetics ascher shapiro deceased institute professor mit contributed development bme field medical devices eg


intra aortic balloons gordana vunjak novakovic university professor columbia university pioneer tissue engineering bioreactor design john g webster professor emeritus university wisconsin madison pioneer field instrumentation amplifiers recording electrophysiological signals fred weibell coauthor ''biomedical instrumentation measurements'' ua whitaker deceased provider whitaker foundation supported research education bme providing million various universities helping create bme programs helping finance construction buildings see also biomedical engineering instrumentation program beip references reading external


german lorenz cipher machine used world war nbsp ii encrypt high level general staff messages '''cryptography''' '''cryptology''' translit krypt hidden secret ''graphein'' write '' logia'' study respectively ref name x sjt ref practice study techniques secure communication presence adversarial behavior ref name rivest ref generally cryptography constructing analyzing protocols prevent third parties public reading private messages ref name modern crypto ref various aspects information security data confidentiality data integrity authentication non repudiation ref name hac ref central modern cryptography modern cryptography exists intersection disciplines

Formal science
Computer science
Computer security
Cryptography


mathematics computer science electrical engineering communication science physics applications cryptography include electronic commerce chip based payment cards digital currencies computer passwords military communications cryptography prior modern age effectively synonymous '''encryption''' converting information readable state unintelligible nonsense sender encrypted message shares decoding technique intended recipients preclude access adversaries cryptography literature often uses names alice sender bob b intended recipient eve eavesdropper adversary ref name codesintro ref since development rotor cipher machines world war nbsp


advent computers world war nbsp ii cryptography methods become increasingly complex applications varied modern cryptography heavily based mathematical theory computer science practice cryptographic algorithms designed around computational hardness assumptions making algorithms hard break actual practice adversary theoretically possible break well designed system infeasible actual practice schemes well designed therefore termed computationally secure theoretical advances eg improvements integer factorization algorithms faster computing technology require designs continually reevaluated


necessary adapted exist information theoretically secure schemes cannot broken even unlimited computing power one time pad schemes much difficult use practice best theoretically breakable computationally secure schemes growth cryptographic technology raised number legal issues information age cryptography 's potential use tool espionage sedition led many governments classify weapon limit even prohibit use export ref name cryptolaw jurisdictions use


cryptography legal laws permit investigators compel disclosure encryption keys documents relevant investigation ref name uk law ref name rangersteve ref cryptography also plays major role digital rights management copyright infringement disputes regard digital media ref name aacs terminology alphabet shift ciphers believed used julius caesar years ago ref name codesintro example words letters alphabet shifted three one direction encrypt three direction decrypt first use term '''cryptograph'''


opposed '''cryptogram''' dates back th century originating gold bug story edgar allan poe ref name aswjp ref ref name mg vi ref modern times cryptography referred almost exclusively encryption process converting ordinary information called '''plaintext''' unintelligible form called '''ciphertext''' ref name kahnbook '''decryption''' reverse words moving unintelligible ciphertext back plaintext '''cipher''' '''cypher''' pair algorithms carry encryption reversing decryption detailed operation cipher controlled algorithm instance key


key secret ideally known communicants usually string characters ideally short remembered user needed decrypt ciphertext formal mathematical terms cryptosystem ordered list elements finite possible plaintexts finite possible cyphertexts finite possible keys encryption decryption algorithms correspond key keys important formally actual practice ciphers without variable keys trivially broken knowledge cipher used therefore useless even counter productive


security communication largely relation two keys hard discover ref name dmb q ref examples asymmetric systems include rsa rivest shamir adleman ecc elliptic curve cryptography quality symmetric algorithms include commonly used aes advanced encryption standard replaced older des data encryption standard ref name yd mw ref high quality symmetric algorithms include assorted children 's language tangling schemes pig latin cant indeed effectively cryptographic schemes however seriously intended source prior invention one time pad early th century colloquial use term


'''code''' often used mean method encryption concealment meaning however cryptography code specific meaning replacement unit plaintext ie meaningful word phrase '''code word''' example wallaby replaces attack dawn cypher contrast scheme changing substituting element level letter syllable pair letters etc order produce cyphertext cryptanalysis term used study methods obtaining meaning encrypted information without


sense advises '' 'steganography''' sometimes included cryptology ref name xngb ref study characteristics languages application cryptography cryptology eg frequency data letter combinations universal patterns etc called '''cryptolinguistics''' history cryptography cryptanalysis modern era cryptography focused message confidentiality ie encryption conversion messages comprehensible form incomprehensible one back end rendering unreadable interceptors eavesdroppers without secret knowledge namely key needed decryption message encryption attempted ensure secrecy communications


spies military leaders diplomats recent decades field expanded beyond confidentiality concerns include techniques message integrity checking sender receiver identity authentication digital signatures interactive proofs secure computation among others classic cryptography reconstructed ancient greek ' 'scytale'' early cipher device main classical cipher types transposition ciphers rearrange order letters message eg 'hello world' becomes 'ehlol owrdl' trivially simple rearrangement scheme substitution ciphers systematically replace letters groups letters letters groups letters eg 'fly once' becomes 'gmz bu podf'


replacing letter one following latin alphabet simple versions either never offered much confidentiality enterprising opponents early substitution cipher caesar cipher letter plaintext replaced letter fixed number positions alphabet suetonius reports julius caesar used shift three communicate generals atbash example early hebrew cipher earliest known use cryptography carved ciphertext stone egypt ca bce may


done amusement literate observers rather way concealing information greeks classical times said known ciphers eg scytale transposition cipher claimed used spartan military ref name jrt g publisher ams bookstore page isbn ref steganography ie hiding even existence message keep confidential also first developed ancient times early example herodotus message tattooed slave 's shaved head concealed regrown hair ref name kahnbook ref modern examples steganography include use invisible ink microdots


digital watermarks conceal information india year old kamasutra v tsy yana speaks two different kinds ciphers called kautiliyam mulavediya kautiliyam cipher letter substitutions based phonetic relations vowels becoming consonants mulavediya cipher alphabet consists pairing letters using reciprocal ones ref name kahnbook sassanid persia two secret scripts according muslim author ibn al nadim '' h dab r ya'' literally king 's script used official correspondence ''r z sahar ya'' used communicate secret messages countries ref name voluq ref david kahn notes


al kindi also known ''alkindus'' th century ref name singh ref nearly ciphers could broken informed attacker classical ciphers still enjoy popularity today though mostly puzzles see cryptogram al kindi wrote book cryptography entitled ''risalah fi istikhraj al mu'amma'' ''manuscript deciphering cryptographic messages'' described first known use frequency analysis cryptanalysis techniques ref name singh ref name kadi ref french cipher machine arms henri ii france enciphered letter gabriel de luetz d'aramon french ambassador ottoman empire partial decipherment language letter frequencies may offer little help


extended historical encryption techniques homophonic cipher tend flatten frequency distribution ciphers language letter group n gram frequencies may provide attack essentially ciphers remained vulnerable cryptanalysis using frequency analysis technique development polyalphabetic cipher clearly leon battista alberti around year though indication already known al kindi ref name kadi alberti 's innovation use different ciphers ie substitution alphabets various parts message perhaps successive plaintext letter limit also invented


probably first automatic cipher device wheel implemented partial realization invention vigen cipher polyalphabetic cipher encryption uses ''key word'' controls letter substitution depending letter key word used mid th century charles babbage showed vigen cipher vulnerable kasiski examination first published ten years later friedrich kasiski although frequency analysis powerful general technique many ciphers encryption still often effective practice many would cryptanalyst unaware


technique breaking message without using frequency analysis essentially required knowledge cipher used perhaps key involved thus making espionage bribery burglary defection etc attractive approaches cryptanalytically uninformed finally explicitly recognized th century secrecy cipher 's algorithm sensible practical safeguard message security fact realized adequate cryptographic scheme including ciphers remain secure even adversary fully understands cipher algorithm security key used alone sufficient good cipher


maintain confidentiality attack fundamental principle first explicitly stated auguste kerckhoffs generally called kerckhoffs 's principle alternatively bluntly restated claude shannon inventor information theory fundamentals theoretical cryptography ''shannon 's maxim'' 'the enemy knows system' different physical devices aids used assist ciphers one earliest may scytale ancient greece rod supposedly used spartans aid transposition cipher medieval times aids invented


cipher grille also used kind steganography invention polyalphabetic ciphers came sophisticated aids alberti 's cipher disk johannes trithemius' tabula recta scheme thomas jefferson 's wheel cypher publicly known reinvented independently bazeries around many mechanical encryption decryption devices invented early th century several patented among rotor machines famously including enigma machine used german government military late world war ii ref name qc il ref ciphers implemented better quality examples machine designs brought


substantial increase cryptanalytic difficulty wwi ref name rcwgz ref computer era prior early th century cryptography mainly concerned linguistic lexicographic patterns since emphasis shifted cryptography makes extensive use mathematics including aspects information theory computational complexity statistics combinatorics abstract algebra number theory finite mathematics generally cryptography also branch engineering unusual one since deals active intelligent malevolent opposition kinds engineering eg civil chemical engineering need deal neutral natural forces also active research examining relationship


cryptographic problems quantum physics development digital computers electronics helped cryptanalysis made possible much complex ciphers furthermore computers allowed encryption kind data representable binary format unlike classical ciphers encrypted written language texts new significant computer use thus supplanted linguistic cryptography cipher design cryptanalysis many computer ciphers characterized operation binary bit sequences sometimes groups blocks unlike classical mechanical schemes generally manipulate traditional characters ie letters digits


directly however computers also assisted cryptanalysis compensated extent increased cipher complexity nonetheless good modern ciphers stayed ahead cryptanalysis typically case use quality cipher efficient ie fast requiring resources memory cpu capability breaking requires effort many orders magnitude larger vastly larger required classical cipher making cryptanalysis inefficient impractical effectively impossible advent modern cryptography cryptanalysis new mechanical devices proved


difficult laborious united kingdom cryptanalytic efforts bletchley park wwii spurred development efficient means carrying repetitious tasks culminated development colossus world 's first fully electronic digital programmable computer assisted decryption ciphers generated german army 's lorenz sz machine extensive open academic research cryptography relatively recent began mid recent times ibm personnel designed algorithm became federal ie us data encryption standard whitfield diffie martin hellman published key agreement algorithm ref name dh


rsa algorithm published martin gardner 's ''scientific american'' column since cryptography become widely used tool communications computer networks computer security generally modern cryptographic techniques keep keys secret certain mathematical problems intractable integer factorization discrete logarithm problems deep connections abstract mathematics cryptosystems proven unconditionally secure one time pad one proven claude shannon important algorithms proven


secure certain assumptions example infeasibility factoring extremely large integers basis believing rsa secure systems even proof unbreakability unavailable since underlying mathematical problem remains open practice widely used believed unbreakable practice competent observers systems similar rsa one michael rabin provably secure provided factoring ''n pq'' impossible quite unusable practice discrete logarithm problem basis believing


cryptosystems secure related less practical systems provably secure relative solvability insolvability discrete log problem ref name ootpg ''cryptography theory practice'' third edition discrete mathematics applications douglas r stinson chapman hall crc ref well aware cryptographic history cryptographic algorithm system designers must also sensibly consider probable future developments working designs instance continuous improvements computer processing power increased scope brute force attacks specifying key lengths required key lengths similarly advancing ref name fortify ref potential effects


quantum computing already considered cryptographic system designers developing post quantum cryptography announced imminence small implementations machines may making need preemptive caution rather merely speculative ref name hac modern cryptography symmetric key cryptography symmetric key cryptography single key used encryption decryption symmetric key cryptography refers encryption methods sender receiver share key less commonly keys different related easily computable way kind encryption publicly known june ref


name dh ref idea cipher used versions pgp openpgp compatible software time efficient encryption messages symmetric key ciphers implemented either block ciphers stream ciphers block cipher enciphers input blocks plaintext opposed individual characters input form used stream cipher data encryption standard des advanced encryption standard aes block cipher designs designated cryptography standards us government though des 's designation finally withdrawn aes adopted ref name aes ref despite deprecation official standard des especially still approved


created based hidden internal state changes cipher operates internal state initially set using secret key material rc widely used stream cipher ref name hac block ciphers used stream ciphers generating blocks keystream place pseudorandom number generator applying xor operation bit plaintext bit keystream message authentication codes macs much like cryptographic hash functions except secret key used authenticate hash value upon receipt ref name hac


hash algorithm series md like hash functions sha flawed algorithm agency withdrew sha widely deployed secure md cryptanalysts identified attacks sha family improves sha vulnerable clashes us standards authority thought prudent security perspective develop new standard significantly improve robustness nist 's overall hash algorithm toolkit ref name vbub br ref thus hash function design competition meant select new us national standard called sha competition ended


october nist announced keccak would new sha hash algorithm ref name dsijp ref unlike block stream ciphers invertible cryptographic hash functions produce hashed output cannot used retrieve original input data cryptographic hash functions used verify authenticity data retrieved untrusted source add layer security public key cryptography public key cryptography different keys used encryption decryption padlock icon firefox web browser indicates tls public key cryptography system use symmetric key cryptosystems use


key encryption decryption message although message group messages different key others significant disadvantage symmetric ciphers key management necessary use securely distinct pair communicating parties must ideally share different key perhaps ciphertext exchanged well number keys required increases square number network members quickly requires complex key management schemes keep consistent secret groundbreaking paper whitfield diffie martin hellman proposed


notion ''public key'' also generally called ''asymmetric key'' cryptography two different mathematically related keys used ''public'' key ''private'' key ref name hfoql ref public key system constructed calculation one key 'private key' computationally infeasible 'public key' even though necessarily related instead keys generated secretly interrelated pair ref name lojlb ralph merkle working similar ideas time encountered publication delays hellman suggested term used diffie hellman merkle aysmmetric key cryptography ref historian david kahn


secretly agree shared encryption key ref name dh x standard defines commonly used format public key certificates ref name ftyi ref diffie hellman 's publication sparked widespread academic efforts finding practical public key encryption system race finally ronald rivest adi shamir len adleman whose solution since become known rsa algorithm ref name xpdbc br br previously released mit technical memo april published martin gardner 's ''scientific american'' mathematical recreations column ref diffie hellman rsa algorithms addition first publicly known examples


attempt detectable digital signature schemes two algorithms one ' 'signing'' secret key used process message hash message one ' 'verification'' matching public key used message check validity signature rsa dsa two popular digital signature schemes digital signatures central operation public key infrastructures many network security schemes eg ssl tls many vpns etc ref name schneierbook public key algorithms often based


hash used example digital signature good hash functions attacker cannot find two messages produce hash md long used hash function broken md strengthened variant md also widely used broken practice us national security agency developed secure hash algorithm series md like hash functions sha flawed algorithm agency withdrew sha widely deployed secure md cryptanalysts identified attacks sha family improves sha vulnerable


clashes us standards authority thought prudent security perspective develop new standard significantly improve robustness nist 's overall hash algorithm toolkit ref name vbub br ref thus hash function design competition meant select new us national standard called sha competition ended october nist announced keccak would new sha hash algorithm ref name dsijp ref unlike block stream ciphers invertible cryptographic hash functions produce hashed output cannot used retrieve original input


data cryptographic hash functions used verify authenticity data retrieved untrusted source add layer security cryptanalysis variants enigma machine used germany 's military civil authorities late world war ii implemented complex electro mechanical polyalphabetic cipher breaking reading enigma cipher poland 's cipher bureau years war subsequent decryption bletchley park important allied victory ref name kahnbook goal cryptanalysis find weakness insecurity cryptographic scheme thus permitting subversion


evasion common misconception every encryption method broken connection wwii work bell labs claude shannon proved one time pad cipher unbreakable provided key material truly random never reused kept secret possible attackers equal greater length message ref name ynszj ref ciphers apart one time pad broken enough computational effort brute force attack amount effort needed may exponentially dependent key size compared effort needed make use


cipher cases effective security could achieved proven effort required ie work factor shannon 's terms beyond ability adversary means must shown efficient method opposed time consuming brute force method found break cipher since proof found date one time pad remains theoretically unbreakable cipher although well implemented one time pad encryption cannot broken traffic analysis still possible wide variety cryptanalytic attacks classified


several ways common distinction turns eve attacker knows capabilities available ciphertext attack eve access ciphertext good modern cryptosystems usually effectively immune ciphertext attacks known plaintext attack eve access ciphertext corresponding plaintext many pairs chosen plaintext attack eve may choose plaintext learn corresponding ciphertext perhaps many times example gardening used british wwii chosen ciphertext attack eve may able ''choose'' ciphertexts learn corresponding


plaintexts ref name hac finally man middle attack eve gets alice sender bob recipient accesses modifies traffic forwards recipient ref name vwmxk ref also important often overwhelmingly mistakes generally design use one protocols involved pozna monument ''center'' polish cryptanalysts whose breaking germany 's enigma machine ciphers beginning altered course world war ii cryptanalysis symmetric key ciphers typically involves looking attacks block ciphers stream ciphers efficient attack could


perfect cipher example simple brute force attack des requires one known plaintext sup sup decryptions trying approximately half possible keys reach point chances better even key sought found may enough assurance linear cryptanalysis attack des requires sup sup known plaintexts corresponding ciphertexts approximately sup sup des operations ref name junod ref considerable improvement brute force attacks public key algorithms based computational difficulty various problems famous


difficulty integer factorization semiprimes difficulty calculating discrete logarithms yet proven solvable polynomial time using classical turing complete computer much public key cryptanalysis concerns designing algorithms p solve problems using technologies quantum computers instance best known algorithms solving elliptic curve based version discrete logarithm much time consuming best known algorithms factoring least problems less equivalent size thus things equal achieve equivalent strength attack resistance


factoring based encryption techniques must use larger keys elliptic curve techniques reason public key cryptosystems based elliptic curves become popular since invention mid pure cryptanalysis uses weaknesses algorithms attacks cryptosystems based actual use algorithms real devices called ' 'side channel attacks'' cryptanalyst access example amount time device took encrypt number plaintexts report error password pin character may able use timing attack break


cipher otherwise resistant analysis attacker might also study pattern length messages derive valuable information known traffic analysis ref name swt ref quite useful alert adversary poor administration cryptosystem permitting short keys make system vulnerable regardless virtues social engineering attacks humans eg bribery extortion blackmail espionage torture usually employed due cost effective feasible perform reasonable amount time compared pure cryptanalysis high margin cryptographic


primitives much theoretical work cryptography concerns cryptographic ''primitives'' algorithms basic cryptographic properties relationship cryptographic problems complicated cryptographic tools built basic primitives primitives provide fundamental properties used develop complex tools called ''cryptosystems'' ''cryptographic protocols'' guarantee one high level security properties note however distinction cryptographic ''primitives'' cryptosystems quite arbitrary example rsa algorithm sometimes considered cryptosystem sometimes primitive typical examples cryptographic primitives include pseudorandom functions one way functions etc cryptosystems one


cryptographic primitives often used develop complex algorithm called cryptographic system ''cryptosystem'' cryptosystems eg el gamal encryption designed provide particular functionality eg public key encryption guaranteeing certain security properties eg chosen plaintext attack cpa security random oracle model cryptosystems use properties underlying cryptographic primitives support system 's security properties distinction primitives cryptosystems somewhat arbitrary sophisticated cryptosystem derived combination several primitive cryptosystems many cases cryptosystem 's structure involves back forth communication among two


parties space eg sender secure message receiver across time eg cryptographically protected backup data cryptosystems sometimes called ''cryptographic protocols'' widely known cryptosystems include rsa encryption schnorr signature el gamal encryption pgp etc complex cryptosystems include electronic cash ref name hplni ref systems signcryption systems etc 'theoretical' cryptosystems include interactive proof systems ref name yssq ref like zero knowledge proofs ref name blaw ref systems secret sharing ref name krguy ref ref name oedvm ref etc lightweight cryptography lightweight cryptography lwc concerns cryptographic algorithms developed strictly constrained environment growth internet things iot spiked research


development lightweight algorithms better suited environment iot environment requires strict constraints power consumption processing power security ref name desup ref algorithms present aes speck examples many lwc algorithms developed achieve standard set national institute standards technology ref name vybze ref applications general ensure secrecy transmission many systems use private key cryptography protect transmitted information public key systems one maintain secrecy without master key large number keys cybersecurity


cryptography used secure communications encrypting websites use encryption via https ref name ucl risky business url https blogsuclacuk infosec applications cryptography url status live archive url https webarchiveorg web https blogsuclacuk infosec applications cryptography archive date february access date december website blogsuclacuk ref end end encryption sender receiver read messages implemented email pretty good privacy secure messaging general signal whatsapp ref name ucl risky business url https blogsuclacuk infosec applications cryptography url status live archive url https webarchiveorg web https blogsuclacuk infosec applications cryptography archive date february access date december website blogsuclacuk ref operating systems use encryption keep passwords secret conceal parts system ensure software updates truly system maker ref name ucl risky business url https blogsuclacuk infosec applications cryptography url status live archive url https webarchiveorg web https blogsuclacuk infosec applications cryptography archive date february access date december website blogsuclacuk ref instead storing plaintext passwords computer systems store


hashes thereof user logs system passes given password cryptographic hash function compares hashed value file manner neither system attacker point access password plaintext ref name ucl risky business url https blogsuclacuk infosec applications cryptography url status live archive url https webarchiveorg web https blogsuclacuk infosec applications cryptography archive date february access date december website blogsuclacuk ref radically encryption sometimes used encrypt one 's entire drive example university college london implemented bitlocker program microsoft render drive data opaque without users logging ref name ucl risky business url https blogsuclacuk infosec applications cryptography url status live archive url https webarchiveorg web https blogsuclacuk infosec applications cryptography archive date february access date december website blogsuclacuk ref electronic money blockchain


cryptocurrency social issues legal issues prohibitions cryptography long interest intelligence gathering law enforcement agencies ref name rangersteve secret communications may criminal even treasonous facilitation privacy diminution privacy attendant prohibition cryptography also considerable interest civil rights supporters accordingly history controversial legal issues surrounding cryptography especially since advent inexpensive computers made widespread access high quality cryptography possible countries even domestic use cryptography restricted


france significantly restricted use cryptography domestically though since relaxed many rules china iran license still required use cryptography ref name cryptolaw ref many countries tight restrictions use cryptography among restrictive laws belarus kazakhstan mongolia pakistan singapore tunisia vietnam ref name whfou ref united states cryptography legal domestic use much conflict legal issues related cryptography ref name rangersteve one particularly important issue export cryptography cryptographic software hardware probably


importance cryptanalysis world war ii expectation cryptography would continue important national security many western governments point strictly regulated export cryptography world war ii illegal us sell distribute encryption technology overseas fact encryption designated auxiliary military equipment put united states munitions list ref name cyberlaw br ref development personal computer asymmetric key algorithms ie public key techniques internet especially problematic however internet grew computers became


widely available high quality encryption techniques became well known around globe export controls several challenges us export regulation cryptography source code philip zimmermann 's pretty good privacy pgp encryption program found way onto internet june complaint rsa security called rsa data security inc resulted lengthy criminal investigation zimmermann us customs service fbi though charges ever filed ref name zim ref ref name levybook ref daniel j bernstein graduate student uc berkeley brought lawsuit us government


challenging aspects restrictions based free speech grounds case bernstein v united states ultimately resulted decision printed source code cryptographic algorithms systems protected free speech united states constitution ref name b v us ref thirty nine countries signed wassenaar arrangement arms control treaty deals export arms dual use technologies cryptography treaty stipulated use cryptography short key lengths bit symmetric encryption bit rsa would longer export controlled ref name wa ref cryptography exports us


became less strictly regulated consequence major relaxation ref name ftoou ref longer many restrictions key sizes us exported mass market software since relaxation us export restrictions personal computers connected internet include us sourced web browsers firefox internet explorer almost every internet user worldwide potential access quality cryptography via browsers eg via transport layer security mozilla thunderbird microsoft outlook e mail client programs similarly transmit receive emails via tls send receive email encrypted mime many


internet users n't realize basic application software contains extensive cryptosystems browsers email programs ubiquitous even governments whose intent regulate civilian use cryptography generally n't find practical much control distribution use cryptography quality even laws force actual enforcement often effectively impossible nsa involvement nsa headquarters fort meade maryland another contentious issue connected cryptography united states influence national security agency cipher development policy ref name rangersteve nsa


involved design des development ibm consideration national bureau standards possible federal standard cryptography ref name cryptogram ref des designed resistant differential cryptanalysis ref name coppersmith des ref powerful general cryptanalytic technique known nsa ibm became publicly known rediscovered late ref name kmod ref according steven levy ibm discovered differential cryptanalysis ref name levybook kept technique secret nsa 's request technique became publicly known biham shamir discovered announced years


later entire affair illustrates difficulty determining resources knowledge attacker might actually another instance nsa 's involvement clipper chip affair encryption microchip intended part capstone cryptography control initiative clipper widely criticized cryptographers two reasons cipher algorithm called skipjack classified declassified long clipper initiative lapsed classified cipher caused concerns nsa deliberately made cipher weak order assist intelligence efforts whole initiative also criticized based violation


kerckhoffs 's principle scheme included special escrow key held government use law enforcement ie wiretapping ref name levybook digital rights management cryptography central digital rights management drm group techniques technologically controlling use copyrighted material widely implemented deployed behest copyright holders us president bill clinton signed digital millennium copyright act dmca criminalized production dissemination use certain cryptanalytic techniques technology known later discovered specifically could used circumvent drm technological schemes ref name dmca ref


noticeable impact cryptography research community since argument made cryptanalytic research violated dmca similar statutes since enacted several countries regions including implementation eu copyright directive similar restrictions called treaties signed world intellectual property organization member states united states department justice fbi enforced dmca rigorously feared law nonetheless remains controversial one niels ferguson well respected cryptography researcher publicly stated release


keys responsible blu ray hd dvd content scrambling discovered released onto internet cases motion picture association america sent numerous dmca takedown notices massive internet backlash ref name aacs ref triggered perceived impact notices fair use free speech forced disclosure encryption keys united kingdom regulation investigatory powers act gives uk police powers force suspects decrypt files hand passwords protect encryption keys failure comply offense right punishable conviction


two year jail sentence five years cases involving national security ref name uk law ref successful prosecutions occurred act first ref name vzk j ref resulted term months' imprisonment ref name jof ref similar forced disclosure laws australia finland france india compel individual suspects investigation hand encryption keys passwords criminal investigation united states federal criminal case united states v fricosu addressed whether search warrant compel person reveal encryption passphrase password ref name ml w ref electronic frontier foundation eff argued


violation protection self incrimination given fifth amendment ref name vxom ref court ruled writs act defendant required produce unencrypted hard drive court ref name qpsui ref many jurisdictions legal status forced disclosure remains unclear fbi apple encryption dispute concerns ability courts united states compel manufacturers' assistance unlocking cell phones whose contents cryptographically protected potential counter measure forced disclosure cryptographic software supports plausible deniability encrypted data indistinguishable unused


james gannon ''stealing secrets telling lies spies codebreakers helped shape twentieth century'' washington dc brassey 's oded goldreich http wwwwisdomweizmannacil oded foc bookhtml ''foundations cryptography'' two volumes cambridge university press '' http wwwcsumdedu jkatz imchtml introduction modern cryptography '' jonathan katz yehuda lindell ''alvin 's secret code'' clifford b hicks children 's novel introduces basic cryptography cryptanalysis ibrahim al kadi origins cryptology arab contributions cryptologia vol april pp nbsp https webarchiveorg web http wwwcryptorubde en paarhtml christof paar jan pelzl http wwwcryptography textbookcom ''understanding cryptography textbook students practitioners'' springer slides online cryptography


lectures information available companion web site accessible introduction practical cryptography non mathematicians ''introduction modern cryptography'' phillip rogaway mihir bellare mathematical introduction theoretical cryptography including reduction based security proofs http wwwcsucdavisedu rogaway classes spring book mainpdf pdf download johann christoph woltag 'coded communications encryption ' r diger wolfrum ed ''max planck encyclopedia public international law'' oxford university press giving overview international law issues regarding cryptography jonathan arbib john dwyer ''discrete mathematics cryptography'' st edition external links http ciphersbyrittercom glossaryhtm crypto glossary dictionary technical cryptography https webarchiveorg web http wwwnsagov kids nsa 's cryptokids https webarchiveorg web http wwwcryptoolorg images ct presentations cryptoolpresentation enpdf overview


'''combinatorics''' area mathematics primarily concerned counting means end obtaining results certain properties finite structures closely related many areas mathematics many applications ranging logic statistical physics evolutionary biology computer science full scope combinatorics universally agreed upon according hj ryser definition subject difficult crosses many mathematical subdivisions insofar area described types problems addresses combinatorics involved

Formal science
Mathematics
Applied mathematics
Combinatorics


''enumeration'' counting specified structures sometimes referred arrangements configurations general sense associated finite systems ''existence'' structures satisfy certain given criteria ''construction'' structures perhaps many ways ''optimization'' finding best structure solution among several possibilities largest smallest satisfying ''optimality criterion'' leon mirsky said combinatorics range linked studies something common yet diverge widely objectives methods degree coherence attained one way


define combinatorics perhaps describe subdivisions problems techniques approach used however also purely historical reasons including including topics combinatorics umbrella although primarily concerned finite systems combinatorial questions techniques extended infinite specifically countable discrete setting combinatorics well known breadth problems tackles combinatorial problems arise many areas pure mathematics notably algebra probability theory topology geometry well many application areas many


combinatorial questions historically considered isolation giving ''ad hoc'' solution problem arising mathematical context later twentieth century however powerful general theoretical methods developed making combinatorics independent branch mathematics right one oldest accessible parts combinatorics graph theory numerous natural connections areas combinatorics used frequently computer science obtain formulas estimates analysis algorithms mathematician studies combinatorics called '''' history example


change ringing six bells one earliest nontrivial results graph theory basic combinatorial concepts enumerative results appeared throughout ancient world th century bce ancient indian physician sushruta asserts sushruta samhita combinations made different tastes taken one time two time etc thus computing sup sup nbsp nbsp possibilities greek historian plutarch discusses argument chrysippus rd century bce hipparchus nd century bce rather delicate enumerative problem later shown related schr der hipparchus numbers earlier


works jj sylvester late th century percy macmahon early th century helped lay foundation enumerative algebraic combinatorics graph theory also enjoyed increase interest time especially connection four color problem second half th century combinatorics enjoyed rapid growth led establishment dozens new journals conferences subject part growth spurred new connections applications fields ranging algebra probability functional analysis number theory etc connections shed


problem enumerative combinatorics twelvefold way provides unified framework counting permutations combinations partitions analytic combinatorics analytic combinatorics concerns enumeration combinatorial structures using tools complex analysis probability theory contrast enumerative combinatorics uses explicit combinatorial formulae generating functions describe results analytic combinatorics aims obtaining asymptotic formulae partition theory plane partition partition theory studies various enumeration asymptotic problems related integer partitions closely related q series special functions orthogonal polynomials originally part number theory


part order theory independent field study number connections parts combinatorics extremal combinatorics extremal combinatorics studies extremal questions set systems types questions addressed case largest possible graph satisfies certain properties example largest triangle free graph '' n'' vertices complete bipartite graph ''k sub nn sub '' often hard even find extremal answer ''f'' ''n'' exactly one give asymptotic estimate ramsey theory another part extremal combinatorics states


sufficiently large configuration contain sort order advanced generalization pigeonhole principle probabilistic combinatorics self avoiding walk square grid graph probabilistic combinatorics questions following type probability certain property random discrete object random graph instance average number triangles random graph probabilistic methods also used determine existence combinatorial objects certain prescribed properties explicit examples might difficult find simply observing probability


randomly selecting object properties greater approach often referred ''the'' probabilistic method proved highly effective applications extremal combinatorics graph theory closely related area study finite markov chains especially combinatorial objects probabilistic tools used estimate mixing time often associated paul erd pioneering work subject probabilistic combinatorics traditionally viewed set tools study problems parts combinatorics however growth applications analyze algorithms computer


science well classical probability additive number theory probabilistic number theory area recently grew become independent field combinatorics algebraic combinatorics young diagram partition algebraic combinatorics area mathematics employs methods abstract algebra notably group theory representation theory various combinatorial contexts conversely applies combinatorial techniques problems algebra algebraic combinatorics continuously expanding scope topics techniques seen area mathematics interaction combinatorial algebraic methods particularly strong significant combinatorics


words thue morse infinite word combinatorics words deals formal languages arose independently within several branches mathematics including number theory group theory probability applications enumerative combinatorics fractal analysis theoretical computer science automata theory linguistics many applications new classical chomsky sch tzenberger hierarchy classes formal grammars perhaps best known result field geometric combinatorics icosahedron geometric combinatorics related convex discrete geometry particular polyhedral combinatorics asks example many faces dimension convex polytope metric


properties polytopes play important role well eg cauchy theorem rigidity convex polytopes special polytopes also considered permutohedra associahedra birkhoff polytopes combinatorial geometry historical name discrete geometry topological combinatorics splitting necklace two cuts combinatorial analogs concepts methods topology used study graph coloring fair division partitions partially ordered sets decision trees necklace problems discrete morse theory confused combinatorial topology older name algebraic topology arithmetic combinatorics arithmetic combinatorics


ideas set theory extremal combinatorics gian carlo rota used name ''continuous combinatorics'' describe geometric probability since many analogies ''counting'' ''measure'' related fields kissing spheres connected coding theory discrete geometry combinatorial optimization combinatorial optimization study optimization discrete combinatorial objects started part combinatorics graph theory viewed branch applied mathematics computer science related operations research algorithm theory computational complexity theory coding theory coding theory started


topological combinatorics viewed outgrowths early discrete geometry combinatorics dynamical systems combinatorial aspects dynamical systems another emerging field dynamical systems defined combinatorial objects see example graph dynamical system combinatorics physics increasing interactions combinatorics physics particularly statistical physics examples include exact solution ising model connection potts model one hand chromatic tutte polynomials hand see also combinatorial biology combinatorial chemistry combinatorial data analysis combinatorial game theory combinatorial


group theory list combinatorics topics phylogenetics polynomial method combinatorics notes references bj rner anders stanley richard p http www mathmitedu rstan papers combpdf ''a combinatorial miscellany'' b na mikl http wwwworldscientificcom worldscibooks ''a walk combinatorics rd edition '' graham ronald l groetschel martin lov sz l szl eds ''handbook combinatorics'' volumes amsterdam nl cambridge elsevier north holland mit press lindner charles c rodger christopher eds ''design theory'' crc press st edition stanley richard p http www mathmitedu rstan ec ''enumerative combinatorics'' volumes cambridge university press van lint jacobus h


wilson richard ''a course combinatorics'' nd edition cambridge university press external links https wwwbritannicacom science combinatorics combinatorial analysis article encyclop dia britannica eleventh edition http mathworldwolframcom combinatoricshtml combinatorics mathworld article many references http wwwmathpagescom home icombinahtm combinatorics ''mathpagescom'' portal http wwwcombinatoricsnet resources hyper hyperbookaspx hyperbook combinatorics collection math articles links http wwwdpmmscamacuk wtg culturespdf two cultures mathematics wt gowers article problem solving vs theory building http wwwmathillinoisedu dwest openp glosshtml glossary terms combinatorics https wwwmatunivieacat slc divers softwarehtml list combinatorics software databases


sequences infinite series well defined limit infinitesimal calculus developed independently late th century isaac newton gottfried wilhelm leibniz today calculus widespread uses science engineering economics mathematics education ''calculus'' denotes courses elementary mathematical analysis mainly devoted study functions limits word ''calculus'' plural ''calculi'' latin word meaning originally small pebble meaning kept medicine ndash see calculus medicine pebbles used counting measuring distance travelled transportation devices use ancient

Formal science
Mathematics
Pure mathematics
Analysis
Calculus


rome meaning word evolved today usually means method computation therefore used naming specific methods calculation related theories propositional calculus ricci calculus calculus variations lambda calculus process calculus history modern calculus developed th century europe isaac newton gottfried wilhelm leibniz independently first publishing around time elements appeared ancient greece china middle east still later medieval europe india ancient archimedes used


method exhaustion calculate area parabola ancient period introduced ideas led integral calculus seem developed ideas rigorous systematic way calculations volume area one goal integral calculus found egyptian moscow papyrus th dynasty nbsp bc formulae simple instructions indication method lack major components age greek mathematics eudoxus nbsp bc used method exhaustion foreshadows concept limit


calculate areas volumes archimedes nbsp bc developed idea inventing heuristics resemble methods integral calculus method exhaustion later discovered independently china liu hui rd century ad order find area circle ref name th century ad zu gengzhi son zu chongzhi established method would later called cavalieri 's principle find volume sphere medieval middle east alhazen th century arab mathematician physicist middle east hasan ibn al haytham latinized alhazen nbsp ce derived


complete theory encompassing components well known western world ''taylor series'' ''infinite series approximations'' however able combine many differing ideas two unifying themes derivative integral show connection two turn calculus great problem solving tool today ref name katz modern pp ref johannes kepler 's work ''stereometrica doliorum'' formed basis integral calculus kepler developed method sum radii calculating area ellipse ref name eb ref significant work treatise


origin kepler 's methods ref name eb written bonaventura cavalieri argued volumes areas computed sums volumes areas infinitesimally thin cross sections ideas similar archimedes' ''the method'' treatise believed lost th century rediscovered early th century would unknown cavalieri cavalieri 's work well respected since methods could lead erroneous results infinitesimal quantities introduced disreputable first formal study calculus brought


together cavalieri 's infinitesimals calculus finite differences developed europe around time pierre de fermat claiming borrowed diophantus introduced concept adequality represented equality infinitesimal error term combination achieved john wallis isaac barrow james gregory latter two proving second fundamental theorem calculus around isaac newton developed use calculus laws motion gravitation product rule chain rule notions higher derivatives taylor series analytic functions used isaac


newton idiosyncratic notation applied solve problems mathematical physics works newton rephrased ideas suit mathematical idiom time replacing calculations infinitesimals equivalent geometrical arguments considered beyond reproach used methods calculus solve problem planetary motion shape surface rotating fluid oblateness earth motion weight sliding cycloid many problems discussed ''principia mathematica'' work developed series expansions functions including fractional


irrational powers clear understood principles taylor series publish discoveries time infinitesimal methods still considered disreputable ref name gottfried wilhelm leibniz first state clearly rules calculus ideas arranged true calculus infinitesimals gottfried wilhelm leibniz originally accused plagiarism newton ref name leib leibniz gottfried wilhelm ''the early mathematical manuscripts leibniz'' cosimo inc p https booksgooglecom bookshl en lr id wpc smc oi fnd pg pa dq gottfried wilhelm leibniz accused plagiarism newton v onepage q plagairism f false online copy ref regarded independent inventor contributor calculus contribution


provide clear set rules working infinitesimal quantities allowing computation second higher derivatives providing product rule chain rule differential integral forms unlike newton leibniz paid lot attention formalism often spending days determining appropriate symbols concepts today leibniz newton usually given credit independently inventing developing calculus newton first apply calculus general physics leibniz developed much notation used calculus today basic insights newton leibniz provided


laws differentiation integration second higher derivatives notion approximating polynomial series newton 's time fundamental theorem calculus known newton leibniz first published results great controversy mathematician therefore country deserved credit newton derived results first later published ''method fluxions'' leibniz published nova methodus pro maximis et minimis first newton claimed leibniz stole ideas unpublished notes newton shared members royal society controversy divided english speaking mathematicians


continental european mathematicians many years detriment english mathematics careful examination papers leibniz newton shows arrived results independently leibniz starting first integration newton differentiation leibniz however gave new discipline name newton called calculus science fluxions since time leibniz newton many mathematicians contributed continuing development calculus one first complete works infinitesimal integral calculus written maria gaetana agnesi


maria gaetana agnesi foundations calculus ''foundations'' refers rigorous development subject axioms definitions early calculus use infinitesimal quantities thought unrigorous fiercely criticized number authors notably michel rolle bishop berkeley berkeley famously described infinitesimals ghosts departed quantities book ''the analyst'' working rigorous foundation calculus occupied mathematicians much century following newton leibniz still extent active area research today several mathematicians including


maclaurin tried prove soundness using infinitesimals would years later due work cauchy weierstrass way finally found avoid mere notions infinitely small quantities foundations differential integral calculus laid cauchy 's ''cours d'analyse'' find broad range foundational approaches including definition continuity terms infinitesimals somewhat imprecise prototype definition limit definition differentiation work weierstrass formalized concept limit eliminated infinitesimals although


definition actually validate nilsquare infinitesimals following work weierstrass eventually became common base calculus limits instead infinitesimal quantities though subject still occasionally called infinitesimal calculus bernhard riemann used ideas give precise definition integral also period ideas calculus generalized euclidean space complex plane modern mathematics foundations calculus included field real analysis contains full definitions proofs theorems calculus reach calculus also


greatly extended henri lebesgue invented measure theory used define integrals pathological functions laurent schwartz introduced distributions used take derivative function whatsoever limits rigorous approach foundation calculus another way use abraham robinson 's non standard analysis robinson 's approach developed uses technical machinery mathematical logic augment real number system infinitesimal infinite numbers original newton leibniz conception resulting numbers called hyperreal numbers used


give leibniz like development usual rules calculus also smooth infinitesimal analysis differs non standard analysis mandates neglecting higher power infinitesimals derivations significance many ideas calculus developed earlier greece china india iraq persia japan use calculus began europe th century isaac newton gottfried wilhelm leibniz built work earlier mathematicians introduce basic principles ref name ref ref name ref development calculus built earlier concepts instantaneous motion area


underneath curves applications differential calculus include computations involving velocity acceleration slope curve optimization applications integral calculus include computations involving area volume arc length center mass work pressure advanced applications include power series fourier series calculus also used gain precise understanding nature space time motion centuries mathematicians philosophers wrestled paradoxes involving division zero sums infinitely many numbers questions arise study motion area ancient greek philosopher zeno elea gave several


famous examples paradoxes calculus provides tools especially limit infinite series resolve paradoxes principles limits infinitesimals calculus usually developed working small quantities historically first method infinitesimals objects treated like real numbers sense infinitely small example infinitesimal number could greater less number sequence thus less positive real number point


view calculus collection techniques manipulating infinitesimals symbols ''' math dx math ''' math dy math taken infinitesimal derivative math dy dx math simply ratio infinitesimal approach fell favor th century difficult make notion infinitesimal precise however concept revived th century introduction non standard analysis smooth infinitesimal analysis provided solid foundations manipulation infinitesimals late th century infinitesimals replaced within academia epsilon delta approach limits limits describe


value function certain input terms values nearby inputs capture small scale behavior context real number system treatment calculus collection techniques manipulating certain limits infinitesimals get replaced small numbers infinitely small behavior function found taking limiting behavior smaller smaller numbers limits thought provide rigorous foundation calculus reason became standard approach twentieth century differential calculus ''f '' ''x'' curve


point slope rise run line tangent curve point differential calculus study definition properties applications derivative function process finding derivative called ' 'differentiation'' given function point domain derivative point way encoding small scale behavior function near point finding derivative function every point domain possible produce new function called ' 'derivative function''


' 'derivative'' original function formal terms derivative linear operator takes function input produces second function output abstract many processes studied elementary algebra functions usually input number output another number example doubling function given input three outputs six squaring function given input three outputs nine derivative however take squaring function input means derivative takes


information squaring function two sent four three sent nine four sent sixteen uses information produce another function function produced deriving squaring function turns doubling function explicit terms doubling function may denoted ''x'' squaring function ''x'' sup sup derivative takes function defined expression input information two sent four three sent nine


four sent sixteen uses information output another function function ''x'' turn common symbol derivative apostrophe like mark called prime thus derivative function called denoted pronounced f prime instance ''x'' sup sup squaring function ''x'' derivative doubling function notation known lagrange 's notation input function represents time derivative represents change respect time example


function takes time input gives position ball time output derivative position changing time velocity ball function linear graph function straight line function written ''mx'' ''b'' independent variable dependent variable ''y'' intercept math frac text rise text run frac text change text change x frac delta delta x math


gives exact value slope straight line graph function straight line however change divided change varies derivatives give exact meaning notion change output respect change input concrete let function fix point domain point graph function number close zero number close therefore


close slope two points math frac f h f h frac f h f h math expression called ' 'difference quotient'' line two points curve called ' 'secant line'' slope secant line secant line approximation behavior function point account happens possible discover behavior setting zero


would require dividing zero undefined derivative defined taking limit tends zero meaning considers behavior small values extracts consistent value case equals zero math lim h f h f h math geometrically derivative slope tangent line graph tangent line limit secant lines derivative limit difference quotients reason derivative sometimes called slope


function particular example derivative squaring function input let ''x'' sup sup squaring function slope note vertical horizontal scales image different math begin align f' lim h h h lim h h h h lim h h h h lim h h end align math slope tangent line squaring function point say


another example correct notation could math begin align g dt g end align math even calculus developed using limits rather infinitesimals common manipulate symbols like real numbers although possible avoid manipulations sometimes notationally convenient expressing operations total derivative integral calculus ''integral calculus'' study definitions properties applications two related concepts ''indefinite integral'' ' 'definite


integral'' process finding value integral called ''integration'' technical language integral calculus studies two related linear operators ''indefinite integral'' also known ''antiderivative'' inverse operation derivative indefinite integral derivative use lower upper case letters function indefinite integral common calculus ' 'definite integral'' inputs function outputs number gives algebraic sum areas graph input x axis technical definition


definite integral involves limit sum areas rectangles called riemann sum motivating example distances traveled given time math mathrm distance mathrm speed cdot mathrm time math speed constant multiplication needed speed changes powerful method finding distance necessary one method approximate distance traveled breaking time many short intervals time multiplying time elapsed interval one speeds interval taking sum


riemann sum approximate distance traveled interval basic idea short time elapses speed stay less however riemann sum gives approximation distance traveled must take limit riemann sums find exact distance traveled constant velocity ''b'' velocity constant total distance traveled given time interval computed multiplying velocity time example travelling steady nbsp mph hours results total distance


miles diagram left constant velocity time graphed two values form rectangle height equal velocity width equal time elapsed therefore product velocity time also calculates rectangular area constant velocity curve connection area curve distance traveled extended ''any'' irregularly shaped region exhibiting fluctuating velocity given time period diagram right represents speed varies time distance traveled


function ''h'' sum rectangles gives approximation area axis curve approximation total distance traveled smaller value give rectangles cases better approximation exact answer need take limit approaches zero symbol integration math int math elongated ''s'' ''s'' stands sum definite integral written math int b f x dx math read integral ''a'' ''b'' ''f'' ''x''


respect ''x'' leibniz notation intended suggest dividing area curve infinite number rectangles width becomes infinitesimally small formulation calculus based limits notation math int b cdots dx math understood operator takes function input gives number area output terminating differential number multiplied although serving reminder limit definition


treated symbolic manipulations integral formally differential indicates variable function integrated serves closing bracket integration operator indefinite integral antiderivative written math int f x dx math functions differing constant derivative shown antiderivative given function actually family functions differing constant since derivative function ''x'' sup sup ''c'' constant ''x'' antiderivative latter given


math int x dx x c math unspecified constant present indefinite integral antiderivative known constant integration fundamental theorem fundamental theorem calculus states differentiation integration inverse operations precisely relates values antiderivatives definite integrals usually easier compute antiderivative apply definition definite integral fundamental theorem calculus provides practical way computing definite integrals also interpreted precise statement fact differentiation


inverse integration fundamental theorem calculus states function continuous interval function whose derivative interval math int b f x dx f b f math furthermore every interval math frac dx int x f dt f x math realization made newton leibniz based results earlier work isaac barrow key proliferation analytic results work became known fundamental theorem provides algebraic method computing many definite integrals without performing limit processes


motion implies derivative saying ''the'' '''change''' ''of momentum body equal resultant force acting body direction'' commonly expressed today force nbsp nbsp mass nbsp nbsp acceleration implies differential calculus acceleration time derivative velocity second time derivative trajectory spatial position starting knowing object accelerating use calculus derive path maxwell 's theory electromagnetism einstein 's theory general relativity also expressed language differential calculus chemistry also uses calculus determining reaction rates radioactive decay


biology population dynamics starts reproduction death rates model population changes calculus used conjunction mathematical disciplines example used linear algebra find best fit linear approximation set points domain used probability theory determine probability continuous random variable assumed density function analytic geometry study graphs functions calculus used find high points low points maxima minima slope concavity inflection points green 's theorem


gives relationship line integral around simple closed curve c double integral plane region bounded c applied instrument known planimeter used calculate area flat surface drawing example used calculate amount area taken irregularly shaped flower bed swimming pool designing layout piece property discrete green 's theorem gives relationship double integral function around simple closed rectangular curve


''c'' linear combination antiderivative 's values corner points along edge curve allows fast calculation sums values rectangular domains example used efficiently calculate sums rectangular domains images order rapidly extract features detect object another algorithm could used summed area table realm medicine calculus used find optimal branching angle blood vessel maximize flow decay laws particular drug 's elimination body


used derive dosing laws nuclear medicine used build models radiation transport targeted tumor therapies economics calculus allows determination maximal profit providing way easily calculate marginal cost marginal revenue calculus also used find approximate solutions equations practice standard way solve differential equations root finding applications examples methods newton 's method fixed point iteration linear approximation instance spacecraft use variation euler method approximate curved


non standard analysis rich enough applied many branches mathematics books articles dedicated solely traditional theorems calculus often go title non standard calculus smooth infinitesimal analysis another reformulation calculus terms infinitesimals based ideas f w lawvere employing methods category theory views functions continuous incapable expressed terms discrete entities one aspect formulation law excluded middle hold


formulation constructive analysis constructive mathematics branch mathematics insists proofs existence number function mathematical object give construction object constructive mathematics also rejects law excluded middle reformulations calculus constructive framework generally part subject constructive analysis see also lists glossary calculus list calculus topics list derivatives integrals alternative calculi list differentiation identities publications calculus table integrals related topics calculus


finite differences calculus polynomials complex analysis differential equation differential geometry ''elementary calculus infinitesimal approach'' discrete calculus fourier series integral equation mathematical analysis multivariable calculus non classical analysis non standard analysis non standard calculus precalculus mathematical education product integral stochastic calculus taylor series references reading books boyer carl benjamin https booksgooglecom booksid klqshuw fnuc printsec frontcover ''the history calculus conceptual development'' hafner dover edition courant richard ''introduction calculus analysis '' edmund landau ''differential integral calculus'' american mathematical society robert adams ''calculus complete course'' albers donald j richard