RSA Encryption

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RSA Encryption

• William Lu

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RSA Background

• Basic technique first discovered in 1973 by
  Clifford Cocks of CESG (part of British GCHQ)
• Invented in 1977 by Ron Rivest, Adi Shamir and
  Len Adleman

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RSA Uses

• Public key encryption
• Digital signatures

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RSA Algorithm

• Generate two large random primes, p and q, of
  approximately the same size
• e.g. for 1024 bit encryption, p and q should be
  about 512 bits each
• Compute n pq and f (p-1)(q-1)
• Choose e where 1lteltf such that gcd(e,f) 1
• Compute d where 1ltdltf such that ed 1 mod f

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RSA Algorithm

• Public key (e,n)
• Private key (d,n)

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Generate Primes

• Get a pseudo random number
• Use Fermats Little Theorem to test for prime
• For prime n and any a, an mod n a
• For composite n and any a, an mod n ? a

• BUT
• If an mod n a, n could be a composite

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Generate Primes

• Does Fermats Little Theorem guarantee primes?
• NO!

• What is it for?
• With enough rounds, n is probably prime
• Much faster then testing with all primes lt n

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Generate Exponents e and d

• For public exponent, e, pick any prime
• Common choices are 3, 17 and 65537 (216 1)
• For secret exponent, d, compute the modular
  inverse of e mod f
• Use Extended Euclidean Algorithm

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Extended Euclidean Algorithm

• To find inverse of e mod n
• Find quotient and remainder of n/e at each step
• Also carry an auxiliary number ui ui-2
  ui-1qi-2 mod n
• Initialize u0 0 and u1 1
• For each step use the previous e as the current n
  and the previous remainder as the current e
• Repeat until e 0 and the auxiliary number is
  the inverse of e mod n

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Extended Euclidean Algorithm
Inverse of 5 mod 72
n e quotient remainder auxiliary
72 5 14 2 0
5 2 2 1 1
2 1 2 0 58
1 0 29
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Encryption/Decryption

• To encrypt message m
• Public key (e,n)
• c me mod n
• To decrypt cipher c
• Private key (d,n)
• m cd mod n

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Encryption/Decryption

• Public key (5,91)
• Private key (29,91)

• To encrypt message 17
• c 175 mod 91
• c 75

• To decrypt cipher 75
• m 7529 mod 91
• m 17

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Signature

• To sign message m
• Private key (d,n)
• m md mod n
• To verify signature
• Public key (e,n)
• m me mod n

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References

• RSA Algorithm
• DI Management Services
• Fermats Little Theorem
• Mathworld
• Extended Euclidean Algorithm
• Wolfgang Stöcher at Profactor Research
• Bill Cherowitzos references at the University of
  Colorado at Denver
• Ph. D (1983) in mathematics at Columbia
  University