# hap8.html#chap8ex5.

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http://www.d.umn.edu/~jgallian/msproject06/c hap8.html#chap8ex5

Naomi Tesar

 developed by Ron L. Rivest, Adi Shamir and Leonard Adleman in A Method for Obtaining Digital Signatures and Public-Key Cryptosystems (1977)  RSA security depends on the difficulty of factoring large numbers

 encryption: converting information to code  decryption: converting code to information  private-key encryption  shared private key used for encryption and decryption  public-key encryption (RSA)  one public key used for encryption and one private key used for decryption

http://msdn.microsoft.com/en-us/library/ff647097.aspx

Person A’s SSN encrypted [person A’s private key] encrypted SSN encrypted [bank’s public key] person A sends message to bank decrypted [bank’s private key] encrypted SSN encrypted SSN message decrypted [person A’s public key] Person A’s SSN encrypted SSN message

 Two integers are relatively prime if they have no common factors other than 1.   Euler’s Totient Function is defined as the number of positive integers less than or equal to that are relatively prime to. 

 Modular arithmetic is the arithmetic of congruences.  Let,, be integers with. Then is congruent to modulo or provided that divides. 

 message:  the numerical message yet to be encrypted  ciphertext:  the encrypted message  public key:  private key:

 choose two distinct primes and and compute  find  choose such that and  determine satisfying

 person A gives the public key to person B  person B converts the message into an integer such that  person B computes  person B sends to person A

 person A recovers by with private key  person A recovers the original message

 let and, then  find and choose  compute to find  let and encrypt  to decrypt,

Define (1) and (2) as the public and private key representations of a message for.

We want to prove that (3) and.(4) To begin, substitute equations (1) and (2) into equations (3) and (4), respectively.

Thus, and. Now, show that.

By the generation of the private key, we are given. By the definition of modular congruence, we know that. (5)

Now, since and are relatively prime, the totient of is. Next, substitute this into expression (5) to obtain.

By the properties of divisors, we now have and, where there is some integer such that.

Since is prime, and therefore,. (6) Now, by the reflexive property of modular arithmetic, we can write,

which gives us. (7) Next, substitute relation (6) into relation (7) to obtain. (8)

Since is prime, any integer that satisfies (8) must either be relatively prime to (case I) or a multiple of (case II).

Case I: is relatively prime to Fermat’s Little Theorem says. Next, we can write or. (9)

Combining (8) and, (9) we obtain. (10)

Case II: is a multiple of If, then for any integer we know that. By modular congruence, we can now write Thus,.

Therefore, for all. Applying the same method for, we obtain. Since and are relatively prime, we know.

By the modular property of symmetry,. (11) Since, there is only one integer that will satisfy relation (11), and thus. (12) Therefore,.

 The RSA Public-Key Cryptosystem allows users to securely send messages and verify the authenticity of these messages using digital signatures.  The RSA Public-Key Cryptosystem works because the public key representation of the message is the inverse of the private key representation of the message.

 The RSA Public-Key Cryptosystem draws from simple principles from number theory and abstract algebra, yet is nearly unbreakable.  RSA security is based on the difficulty factoring large numbers.  Given with ~1000 digits, it is difficult to recover prime factors and for.

Hungerford, Thomas W. "Chapter 2: Congruence in Z and Modular Arithmetic." Abstract Algebra: An Introduction. Philadelphia: Saunders College, 1990. 23-37. Print. Hungerford, Thomas W. "Chapter 12: Public-Key Cryptography." Abstract Algebra: An Introduction. Philadelphia: Saunders College, 1990. 373-378. Print. Joel Chan, “Three Guys and a Large Number,” Math Horizons 2(3), 1995. Neal Koblitz and Alfred Menezes, “A Survey of Public-Key Cryptosystems,” SIAM Review 46(4), 2004. Robert Boyer and J Strother Moore, “Proof Checking the RSA Public Key Encryption Algorithm,” American Mathematical Monthly 91(3), 1984. R. Rivest, A. Shamir, L. Adleman, “A Method for Obtaining Digital Signatures and Public Key Cryptosystems” MIT/LCS/TM-82, Apr 1977 Weisstein, Eric W. “Euclidean Algorithm.” From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/EuclideanAlgorithm.html Weisstein, Eric W. “Relatively Prime.” From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/RelativelyPrime.html Weisstein, Eric W. “Totient Function.” From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/TotientFunction.html