The Mathematics of RSA


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1 The Mathematics of RSA Dimitri Papaioannou May 24, Introduction Cryptographic systems come in two flavors. Symmetric or Private key encryption and Asymmetric or Public key encryption. Strictly speaking, private key encryption does not need to be symmetric but most current implementations of it are in the sense that the same key is used for encryption and decryption and the algorithm used for the two operations is essentially the same. In public key encryption you have different encryption and decryption key. The encryption key (identified as the public key) can be advertised yet the algorithm guarantees that no one but the owner of the decryption (or private) key will able to decrypt the encrypted messages. This means that the public key cannot be reverse engineered to produce the private key even if the details of the algorithm used are known. To accomplish this, encryption algorithms like the RSA exploit NPhard problems from Number Theory such as the factorization of large integers. 2 Encryption systems 2.1 Early Encryption One of the earliest cryptography systems was the shift cipher (often attributed to Julius Caesar) that works like this: Map the letters of the Alphabet A,,C,... to the numbers 0 to 25. Choose as key an integer k from 1 to 25. Encode each letter by e = a + k mod 26 To decode the message, just compute: a = e k mod 26 for each letter. This cipher is ridiculously easy to crack since there are only 25 possible keys. 1
2 2.2 Private Key Encryption In practice, any encryption system that attempts to decrypt one letter at a time is pretty easy to break. The best you can do is choose a random permutation of all the letters in the alphabet. Even though the space of possible encryptions is large (for English 26! = ), the system can be broken by counting the frequency that each letter appears. A slightly more sophisticated encryption scheme is to encode each letter based on its position. This scheme, which can be found occasionally in modern applications, is also vulnerable to the same method of attack. Modern cryptographic systems break up a message in chunks and encode each chunk. The messages are not encoded with letters of the alphabet but in binary. The DES encryption usually uses chunks that are from 64 to 128 bits and uses a key of equal length to shuffle around the original bits. DES is a computationally efficient algorithm so that large messages can be encrypted and decrypted very fast, especially if it is implemented in hardware. 2.3 Public Key Encryption Public key encryption is an asymmetric encryption system. Suppose that there exists function f, that is easy to compute but is computationally infeasible to compute its inverse. Then a message m encrypted as f(m) can only be decrypted by the holder of the inverse. This is the essence of public key cryptography. The following algorithm guarantees encryption and authentication of the sender. ob wants send message m to Alice. 1. He applies his private function f 1 (m) 2. Then he applies Alice s public function f A to the result to get f A (f 1 (m)). 3. Upon reception of the message, Alice applies her private function f 1 A to decrypt the message and get back f 1 (m). 4. Finally she authenticates the message by applying ob s public function f. 3 RSA The RSA algorithm implements the idea outlined above by using the fact that factorization of large integers is though to be, up to now, an NP hard problem. The algorithm goes as follows: 1. Choose large primes p and q and compute n = pq. 2
3 2. Find a number e relatively prime to (p 1)(q 1). 3. Compute d such that ed = 1 mod (p 1)(q 1). (e, n) is the public key. We feel safe in broadcasting n because we assume that factoring it is computationally infeasible. (d, n) is the private key. To encrypt a message m: 4. Compute c = m e mod n To decrypt the message 5. Compute m = c d mod n At this point it is not obvious why the decryption works. efore embarking on the proof, I will discuss some practical considerations of implementing RSA. 3.1 Practical Considerations Speed of encryption/decryption It must be noted that RSA, and asymmetric encryption in general, is much more computationally expensive than symmetric encryption. This is why in practice RSA is not used to transmit the whole message but rather than to transmit a symmetric encryption key which is in turn used for the encryption of the message. The key is discarded after each communication, there is very little space for attacks that attempt to discover the key. Prime Number Generation The ability to generate large primes is essential for an RSA implementation. The method for accomplishing this is not straightforward but it is well researched and refined in practice. First, a random number generator to generate large odd integers, then a number of primality tests is performed to discard any nonprimes. Any number that survives the tests is prime with very high probability. Computing the public and private key Once you have chosen e, computing d is easy by using the Euclidean Algorithm, an algorithm for integer division. 3.2 Why the RSA algorithm works To prove that RSA works we need some results from Number Theory or Group Theory. I will follow the number theory route that requires less terminology, although it does not establish the results in their full generality. We define Euler s Totient function φ(n) to be the number of integers less than n that are relatively prime to n. 3
4 Lemma 3.1 (Euler s Theorem) If gcd(a, n) = 1, then a φ(n) mod n = 1. Proof. Let U(n) be the set of numbers x such that gcd(x, n) = 1. Consider the set au(n) := {ax mod n : x U(n)}. U(n) and au(n) have the same number of elements. Furthermore, each member of au(n) is also a member of U(n) (since gcd(ax, n) = 1 if gcd(a, x) and gcd(x, n) = 1). So the two sets are equal. Since the two sets are equal, the product of their elements (mod n) are also equal. Label the elements in U(n) as x 1, x 2,..., x φ(n). Then by the above observation ax 1 ax 2... ax φ(n) = x 1 x 2... x φ(n) mod n = a φ(n) = 1 mod n Lemma 3.2 If p and q are primes, then φ(p q) = (p 1)(q 1) Proof. There are pq numbers from 1 to pq inclusive. All the multiples of p (p, 2p, 3p,..., qp) are not relatively prime to pq. The same holds for the multiples of q (q, 2q,..., pq). It can be easily seen that these are the only numbers that are not relatively prime to pq. So the total number of relatively prime numbers is pq p q + 1 = (p 1)(q 1). where we subtract the q multiples of p, the p multiples of q, and add 1 because we double counted pq. Theorem 3.3 The decryption calculation retrieves the original message. Proof.. c d mod n = (m e ) d mod n = m ed mod n We know that ed = 1 mod φ(pq) so there is a number k such that ed = 1+k(p 1)(q 1). m 1+k(p 1)(q 1) = m m φ(n) mod n. ut we have by lemma 3.1 that m φ(n) mod n = 1, from which the theorem follows. 4
5 References [1] Wade Trappe, Lawrence C. Washington, Introduction to Cryptography with Coding Theory, Prentice Hall. [2] Joseph A. Gallian, Contemporary Abstract Algebra third edition, Oxford University Press, 1992 [3] Paulo Ribenboim, The Little ook of ig Primes, SpringerVerlag 5
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