Ciphers, Security and Number Theory : A look into RSA

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Ciphers, Security and Number Theory A look
into RSA

• By Mavis Mather

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Problem Statement

• RSA, named for Ronald L. Rivest, Adi Shamir, and
  Leonard Adleman, is one of the most popular
  ciphers.
• Does RSA live up to its reputation as one of the
  best ciphers?

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What is Cryptography?

• Cryptography
• The science of transforming information into an
  intermediate form which secures that information
  while in storage or in transit

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Terminology

• Plaintext The original, readable message.
• Ciphertext The text that has been encrypted
  with a cipher
• Encryption/Encipherment The process that
  changes plaintext to ciphertext
• Decryption/Decipherment The process that
  changes ciphertext to plaintext
• Key A string of bits that determines the
  mapping of the plaintext to the ciphertext.

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Public Key Algorithms

• Public-key ciphers have two keys, the
  enciphering/public key and the deciphering/secret
  key.
• Two attributes
• Computability
• Public and secret keys are independent

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Why a pair of keys?

• The owner of the private key can give the public
  key to everyone without fearing they could
  discover the private key
• A message sent to the owner, encrypted with the
  owner's public key can be decrypted only by the
  owner with the private key
• A message sent from the owner encrypted with the
  private key can be decrypted by everyone with the
  public key, but only the owner could have sent it
• A message encrypted with the sender's private key
  and the recipient's public key could have come
  only from the sender (authenticity) and can be
  decrypted only by the recipient (secrecy)

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Number Theory

• Number theory is a division of mathematics that
  studies the properties of integers and all
  numbers in general.
• Leads to more exotic notions of number.
• Cryptography is heavily based on number theory.
  

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Euler's Totient function

• Definition The Totient function, represented by
  f(n), is defined as the number of positive
  integers less than n which are relatively prime
  to n.
• Example Take the number 14 f(14) would then
  start with 1,2,3,4,5,6,7,8,9,10,11,12,13.
  Cancel the multiples of 2, and 7. Now we have
  1,3,5,9,11,13 6 relatively prime numbers.

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Eulers Totient function cont

• Theorem f(n) n ?pn (1-1?p)
• Explanation
• Lemma 1 f(pk) pk -pk-1
• Lemma 2 f(mn) f(m)?f(n) (gcd (m,n) 1)
• This theorem can be proved by induction on the
  above two lemmas.
• Example
• f(12) f(2x2x3) 12 (1- 1?2) (1- 1?3) 4

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Fermat's Little Theorem

• Theorem ap ? a (mod p)
• Explanation If a prime divides the product of
  numbers, than it must divide one of the original
  numbers.
• Example Let a2, p5 25 32 2 mod 5

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Totient Theorem

• Theorem
• If gcd(a,n)1 and a lt n, then af(n) 1 (mod n)
• Explanation The Totient theorem is a
  generalized version of Fermats Little Theorem
  that utilizes the Totient function formula.
• n need not be prime.

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RSA

• Take two large primes, p and q, compute their
  product n.
• Choose a number, e, less than n, and relatively
  prime to (p-1)(q-1).
• Find another number d such that (ed-1) is
  divisible by (p-1)(q-1).
• The public key is the pair (n, e) the private
  key is (n, d).

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Encryption and Decryption

• Let E(T) be the encryption process that changes
  plaintext (T) to ciphertext (X). I.e.
• E(T) Tp mod n X
• The decryption process would then be
• D(X)Xq mod n T
• Encryption and Decryption are inverses
• D(E(T)) (Tp mod n)q mod n (X)q mod n T

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Why does this work?

• I need to prove that D(E(T)) T mod n.
• D(E(T)) (Tp mod n)q mod n
• D(E(T)) Tpq mod n
• Fact pq kf(n) 1
• D(E(T)) Tkf(n)1 mod n
• D(E(T)) (Tf(n))k T (mod n)
• But Tf(n) 1 (mod n) from Totient Theorem
• D(E(T)) 1k T (mod n)
• D(E(T)) 1 T (mod n)
• D(E(T)) T (mod n)

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Facts about RSA

• As stated above breaking the code requires
  factoring large numbers.
• The RSA factoring Challenge
• On a 90 MHz Pentium, RSA Data Security's
  cryptographic toolkit BSAFE 3.0 has a throughput
  for private-key operations of 21.6 Kbits per
  second with a 512-bit modulus and 7.4 Kbits per
  second with a 1024-bit modulus.
• fastest RSA hardware has a throughput greater
  than 300 Kbits per second with a 512-bit modulus,
  implying that it performs over 500 RSA
  private-key operations per second.

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Conclusion

• The RSA algorithm is part of many official
  standards worldwide.
• Technology using the RSA algorithm is licensed by
  over 700 companies.
• The estimated installed base of RSA BSAFE
  encryption technologies is around 500 million
• RSA is by far the most widely used public-key
  algorithm in the world.

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RSA-155

• 10941738641570527421809707322040357612003732945449
  2059909138421314763499842889 347847179972578912673
  32497625752899781833797076537244027146743531593354
  333897
• 102639592829741105772054196573991675900716567808
  038066803341933521790711307779
• times 10660348838016845482092722036001287867920795
  8575989291522270608237193062808643

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Questions?