Applet Link http://www.d.umn.edu/~jgallian/msproject06/chap8.html#chap8ex5
An Introduction to the RSA Public-Key Cryptosystem Naomi Tesar
RSA Public-Key Cryptosystem • developed by Ron L. Rivest, AdiShamir and Leonard Adleman inA Method for Obtaining Digital Signatures and Public-Key Cryptosystems (1977) • RSA security depends on the difficulty of factoring large numbers
Encryption/Decryption • 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
Encryption/Decryption http://msdn.microsoft.com/en-us/library/ff647097.aspx
Digital Signatures decrypted [bank’s private key] Person A’s SSN encrypted SSN message encrypted SSN encrypted [person A’s private key] person A sends message to bank encrypted SSN decrypted [person A’s public key] encrypted SSN Person A’s SSN encrypted [bank’s public key] message
Mathematics Involved • 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 .
Mathematics Involved • Modular arithmetic is the arithmetic of congruences. • Let , , be integers with . Then is congruent to modulo or provided that divides .
Definitions • message: • the numerical message yet to be encrypted • ciphertext: • the encrypted message • public key: • private key:
Step 1: Key Generation • choose two distinct primes and and compute • find • choose such that and • determine satisfying
Step 2: Encryption • 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
Step 3: Decryption • person A recovers by with private key • person A recovers the original message
RSA Example • let and , then • find and choose • compute to find • let and encrypt • to decrypt,
Proof Define (1) and (2) as the public and private key representations of a message for .
Proof We want to prove that (3) and . (4) To begin, substitute equations (1) and (2) into equations (3) and (4), respectively.
Proof Thus, and . Now, show that .
Proof By the generation of the private key , we are given . By the definition of modular congruence, we know that . (5)
Proof Now, since and are relatively prime, the totient of is . Next, substitute this into expression (5)to obtain .
Proof By the properties of divisors, we now have and , where there is some integer such that .
Proof Since is prime, and therefore, . (6) Now, by the reflexive property of modular arithmetic, we can write ,
Proof which gives us . (7) Next, substitute relation (6) into relation (7) to obtain . (8)
Proof Since is prime, any integer that satisfies (8) must either be relatively prime to (case I) or a multiple of (case II).
Proof Case I: is relatively prime to Fermat’s Little Theorem says . Next, we can write or . (9)
Proof Combining (8) and , (9) we obtain . (10)
Proof Case II: is a multiple of If , then for any integer we know that . By modular congruence, we can now write Thus, .
Proof Therefore, for all . Applying the same method for , we obtain . Since and are relatively prime, we know .
Proof By the modular property of symmetry, . (11) Since , there is only one integer that will satisfy relation (11), and thus . (12) Therefore, .
Conclusions • 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.
Conclusions • 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 .
References 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
