RSA

1
RSA

• The algorithm was publicly described in 1977 by
  Ron Rivest, Adi Shamir, and Leonard Adleman at
  MIT
• Partly used for PGP (Pretty Good Privacy) to
  encrypt session keys

2
RSA Step 1

• Choose two distinct large random prime numbers p
  and q,
• e.g. p 17 and q 11
• Let n pq, e.g. n1711187
• Choose e such that
• and e and are coprime
• is Eulers totient

3
Eulers Totient

• the totient f(n) of a positive integer n is
  defined to be the number of positive integers
  less than or equal to n that are coprime to n.
• f(9)6 because the six numbers 1, 2, 4, 5, 7 and
  8 are coprime to 9

4
RSA Step 2

• e.g. e 7 thus 7 and (17-1)(11-1) 160 are
  coprime
• e can be published as the public-key exponent
• n is the modulus
• This is all that is needed to encrypt

5
RSA - Encryption

• A cyphertext C can be created from a message M
  using the formulaC Me(mod n)
• Example Message M is X in ASCII 1011000 or 88
  in decimal
• C 887(mod 187) 11

6

• 887(mod 187) 884(mod 187) 882(mod
  187)881(mod 187)
• 881 88 88(mod 187)
• 882 7,744 77(mod 187)
• 884 59,969,536 132(mod 187)
• 887 881 882 884 8877132 894,432
  11(mod 187)

7
RSA - Decryption

• Modulus operation is a one way function
• Given only the public-key (7,187) the only way to
  decrypt is through brute-force i.e. try all
  possible keys
• This problem is simplified because you know how
  the private-key is created.

8
RSA- Private-key

• The decryption key d is created with the
  following formulaed 1(mod(p-1)(q-1))
• e.g 7d1(mod(1610))7d 1 (mod 160)d 23
  using Euclids algorithm

9
Extended Euclids Algorithm

• e d (mod f(n) ) 1 In other words, there is
  another number also relatively prime to f(n) that
  is its reciprocal.
• ax by gcd(a,b)The extended Euclidean
  algorithm is particularly useful when a and b are
  coprime, since x is the modular multiplicative
  inverse of a modulo b

10
Extended Euclids Algorithm

• function extended_gcd(a, b)
• if a mod b 0 return 0, 1
• else x, y extended_gcd(b, a mod b) return
  y, x-y(a div b)
• Example
• extended_gcd(160, 7 ) (-1,1122)
• extended_gcd(7, 160(mod 7) ) (1,0-11)
• extended_gcd(6, 7(mod 6) ) (0,1)
• X -1 Y23
• 23 is the multiplicative inverse of e

11
RSA - Decryption

• Now we have our private-key (d,n) e.g. (23,187)
• M Cd (mod n)M 1123 (mod 187)M 111 (mod
  187)112 (mod 187)114 (mod 187)1116 (mod
  187)(mod 187)M 1112155154 (mod 187)M 88
  X

12
RSA - Cryptanalysis

• The security of RSA is based on two problems
• The problem of factoring large numbers
• The RSA problem

13
Factoring Large Numbers

• RSA-200 is largest number factored so far. It has
  200 decimal digits which corresponds to 663 bits
• The sieving effort is estimated to have taken the
  equivalent of 55 years on a single 2.2 GHz
  Opteron CPU.
• The matrix step reportedly took about 3 months on
  a cluster of 80 2.2 GHz Opterons. The sieving
  began in late 2003 and the matrix step was
  completed in May 2005.

14
RSA Problem

• The RSA problem is the task of finding eth roots
  modulo a composite number N whose factors are not
  known
• In other words to find integer P such that Pe C
  (mod N), given integers N, e and C such that N is
  the product of two large primes, 2 lt e lt N is
  coprime to f(N), and 0 lt C lt N. C is chosen
  randomly within that range
• the most efficient means known to solve the RSA
  problem is to factor the modulus N and thus
  discover the private key