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Primes: a quick tour to spplications and challenges!
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Primes: a quick tour to spplications and challenges!






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Primes: a quick tour to spplications and challenges! Primes: a quick tour to spplications and challenges! Presentation Transcript

  • Prime Numbers: Theory and theirapplication in Public KeyCryptographyAshutosh Tripathi10UCS011
  • A prime number is any natural numbergreater than 1, which cannot be factoredas the product of two smaller numbers.Examples: 2,3,5,7,11,13,17,…,991,…,99991,…,9999991,…
  • The ConvergenceThere are 8 primes in first 20 countingnumbersOnly 4 in next 20Does this sequence ultimately converge to0?The answer dates itself back to ~300 BC
  • Euclid’s TheoremThere are infinitely many primesThe proof to this is one of the earliestappearance of Reductio-ad-absurdum(proof by contradiction)Every natural number greater than1 can beuniquely factored into one or more primesFundamental Theorem of Arithmatic
  • The Prime Number TheoremGauss noticed, as Euclid did, that the primenumbers begin to dwindle out as we gethigher and higher up the number ladder…After lots of calculating and trial and error,Gauss showed that:Density of Primes ≈ 1/log nMore Precisely,limlim ππ(x)(x) = 1= 1xx →→ ∝∝x/ln xx/ln x
  • Few Interesting Results OnPrimesThe Largest Prime is 257,885,161− 1courtesy, GIMPS projectThe Twin Prime ConjectureQuartet for the End of Time. Olivier MessiaenVinogradov Theorem: All odd numbers(sufficiently large can be expressed as sum ofthree primes) Odd and Even GoldbachconjecturesAll large primes have 1,3,7,9 as their last digit
  • Randomness in PrimesGiven n, there is no way we could tell the nthprime, unlike the squares or even Fibonacci seq.Although they are deterministic, but they seem toappear randomly on the number line“God may not play dice with the universe, butsomething strange is going on with the primenumbers” –Paul ErdosThis randomness aids in many modern daycryptographic techniques which rely solely on the“difficulties underlying prime factorization”
  • Fermat’s Little TheoremIf p is a prime then for every 0<a<pap-1≡ 1 (mod p)Proof?All a.i(mod p) are distinct and range between 1and p-1 (inclusive)Fermat’s Test: for any N pick a<N randomlyif aN-1≡ 1 (mod N) implies N is primeCarmichael numbers: numbers that are not primeyet fool the Fermats Test. Ex., 561= 3.11.17
  • Private Key SchemesTwo parties secretly agree on some secret codeEach message therefrom is encoded by firstprocessing it with the secret code (XOR op.)On the receivers End, the same computation isrepeated to recover the original codex ≡ secret codeg ≡ message to be encodedSender’s End e(g) = (g o x)Receivers End d(e(g)) = ((g o x) o x) = g (voila!)
  • The RSA Algorithmby Rivest, Shamir & Adleman of MIT in1977best known & widely used public-keyscheme uses large integers (e.g., 1024 bits)security due to cost of factoring largenumbersOwes its reliability to prime factorization
  • The RSA Algorithm (… contd)Unlike the previous protocol, the RSA scheme isan example of public-key cryptography: anybodycan send a message to anybody else using publiclyavailable information, rather like addresses orphone numbers. Each person has a public keyknown to the whole world and a secret key knownonly to him- or herself. When Alice wants to sendmessage x to Bob, she encodes it using his publickey. He decrypts it using his secret key, to retrievex . Eve is welcome to see as many encryptedmessages for Bob as she likes, but she will not beable to decode them, under certain simpleassumptions.
  • The RSA Algorithm (… contd)Property Pick any two primes p and q and let N =p.q . For any e relatively prime to (p − 1)(q − 1) :1. The mapping x→ xe(mod N) is a bijection on {0, 1,2, ... N-1}2. Moreover the inverse mapping is easily realised,let d be the inverse of e(modulo (p-1)(q-1))then for all x e {0,1,…, N-1}(xe)d= x mod N.
  • The RSA Algorithm (… contd)The first property tells us that the mapping x 7→xe mod N is a reasonable way to encode messagesx ; no information is lost. So, if Bob publishes (N,e) as his public key, everyone else can use it tosend him encrypted messages. The second propertythen tells us how decryption can be achieved. Bobshould retain the value d as his secret key, withwhich he can decode all messages that come to himby simply raising them to the d th power modulo N.
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