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UNIVERSITY OF BIELSKO-BIALA

UNIVERSITY OF BIELSKO-BIALA. AKADEMIA TECHNICZNO-HUMANISTYCZNA. Faculty of Mechanical Engineering and Computer Science. Safety in Information Technology ( Prof . dr hab. inż . Mikołaj Karpiński ). Subject :. Asymmetric Cryptography – RSA ( R ivest , S hamir , A dleman ).

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UNIVERSITY OF BIELSKO-BIALA

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  1. UNIVERSITY OF BIELSKO-BIALA AKADEMIA TECHNICZNO-HUMANISTYCZNA Faculty of Mechanical Engineering and Computer Science www.ath.bielsko.pl

  2. Safety in Information Technology(Prof. dr hab. inż. MikołajKarpiński) Subject: AsymmetricCryptography– RSA (Rivest, Shamir, Adleman) Editor: Georg Schön, 10.11.2011 www.ath.bielsko.pl

  3. Problems withsymmetriccryptography:(Managmentanddistributionofkeys) Sender andrecipientneedtoexchangesecretkey. n participantsrequiren(n −1)/2 keys(6* 10^8 user in 2002 meansapprox. 1,8*10^17 keys) Central distributorindicates high effortandisinsecurewithresprecttotrustworthyness (knowseverything) Whyasymmetriccryptography? Public-key procedure!! ( only decription key or private key needs to be secure) >> to find the private key out of the public key is impossible (state of the art – but quantum computers?). Georg Schön(University of Erlangen - Nürnberg)

  4. U U E E Asymmetriccommunication !Public keys are accessible for everyone! Message transfer Decripts with his private key Bob Encrypts with Bob´s public key Alice Georg Schön(University of Erlangen - Nürnberg)

  5. Public keyindex NamePublic key Bob 13121311235912753192375134123 Paul 84228349645098236102631135768 Alice 54628291982624638121025032510 Bob Alice No secure keys for the exchange necessary!But:How to make sure the public key is not replaced by a third person?>> (Public key indexes use digital signatures!) Georg Schön(University of Erlangen - Nürnberg)

  6. RSA cipher • Inventedby Ron Rivest, Adi Shamirand Len Adleman • Ist securitymakesuseofthedifficultytodecompound large numbers in prime factors! A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37…) Georg Schön(University of Erlangen - Nürnberg)

  7. Prime multiplication Bit length: 768 Decimal length: 232 Current PCs can quickly factor numbers with about “80 digits”. Therefore, practical RSA implementations must use moduli with at least “300 digits” to achieve sufficient security! Georg Schön(University of Erlangen - Nürnberg)

  8. Mathematicbackground 1. The modulooperator 2. Euler´stotientfunction 3. Euler-Fermat theorem Rest Divisor Georg Schön(University of Erlangen - Nürnberg)

  9. Euler’s totient function φ of an integer returns how many positive integers a are coprime and smaller than N. Euler´stotientfunction Phi of N is the quantity of positive integers awhere: Georg Schön(University of Erlangen - Nürnberg)

  10. Euler-Fermat theorem • Is a cyclicfunction (resultsrepeatthemselves) • Example: N = 10 a = 3 >>>>> a = 7 >>>>> Nofurtherexplanation. Georg Schön(University of Erlangen - Nürnberg)

  11. Key generation Choose two primes and with Calculate their product: Calculate the value of Euler’s totient function of >>>>> 3 and 7 >>>>> 21 = 3*7 >>>>> 12 = (3-1)*(7-1) Determine D and E: D*E 1 mod 12(eg. Compound number 1, 13, 25, 37, 49, 61, 73, 85, ...)85 = 5 * 17 (D=5, E=17) (N,E – private key; N,D – public key) For defining D, E also see:extended Euclidean algorithm! Georg Schön(University of Erlangen - Nürnberg)

  12. Encryption/Decryption The messagethatistobe send, shallbe9 The userwithkeyE (asencrypt) reckons:9E=95=59049 18 mod 21Sender transmitsencryptedmessage (18) tothereceiver, whouseshis private keyDtodecryptthemessageandreckons:18D=1817=2185911559738696531968 9 mod 21 (originmessage) Georg Schön(University of Erlangen - Nürnberg)

  13. Safety in Information Technology(Prof. dr hab. inż. MikołajKarpiński) ThanksforyourAttention!AnyQuestions? www.ath.bielsko.pl

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