## RSA AlgorithmWe talked about the ideas behind public key cryptography last lecture. How do we implement such a scheme? This is where number theory comes to the rescue in the form of the RSA crypto system. RSA stands for Rivest, Shamir and Adelman, who discovered the scheme in 1977. Clifford Cocks had independently discovered this earlier in 1973, but his work was classified and remained unknown for many years. A link to the original RSA paper is here. ## Messages As NumbersFirst, let us get some preliminary concepts out of the way. We will regard messages as numbers. The idea is that your message is encoded as a number through a scheme such as ASCII. The rest of this presentation will deal with encrypting and decrypting numbers. ## Basic IdeaSuppose Alice wished to send a message to Bob that she wished Bob and no one else to read. In a public key system, she will obtain Bob's public key and encrypt the message using Bob's public key to obtain a encrypted message . This is sent to Bob. Upon receiving the message from Alice, Bob decrypts it using his private key. No one else can decrypt the message unless they have Bob's private key. How do we implement this in practice? The basic idea behind RSA is to create a The reason is called Given and , it is easy to compute to encrypt . However, someone who is snooping the channel sees and . It is computationally hard to invert and compute the original message .
However, if it is hard for someone to invert , how is it that Bob can do so? We will assume that using the private key, Bob has a function that will allow him to compute .
So our scheme should give us functions where is easy to compute but very hard to invert. On the other hand, is easy to compute and inverts if the private key is available. ## Number Theory to the RescueThe basic scheme for RSA uses a really large number . The public key is a pair of numbers The private key is a pair of numbers .
Each message is assumed to be a number between and . If is really large, we allow a large space of numbers to code our messages with. The basic idea is to encrypt a message by computing . This operation is called Example
Let us take a message and encrypt it using the public key . We will simply compute On the first sight, this modular exponentiation looks like an
atrociously hard computation. But remember that is
Therefore, we can first compute this by repeated squaring and modulo operations. Here is how we compute for our example using python interpreter Example Computation
bash$ python Python 2.7.6 (v2.7.6:3a1db0d2747e, Nov 10 2013, 00:42:54) [GCC 4.2.1 (Apple Inc. build 5666) (dot 3)] on darwin Type "help", "copyright", "credits" or "license" for more information. >>> c1 = (1098 **2) % 41989 >>> c1 29912 >>> c2 = (c1 **2) % 41989 >>> c2 26132 >>> c3 = (c2 **2) %41989 >>> c3 14317 >>> c4 = (c3 * c2) % 41989 >>> c4 9854 >>> c5 = (c4 * 1098) % 41989 >>> c5 28519 However, a snooper who simply sees the encrypted message and the
public key will find it very hard to figure out the message
. This problem is considered to be a computationally hard problem,
and is called the So far, we have identified our one way function ,
which is given by Now the idea is to find a private key that satisfies . In other words, if we take the encrypted message and perform modular exponentiation with the private key , we obtain the original message back. This means that we need to satisfy the property that for any message , In other words, we require that . The question is how can we find a pair that will satisfy this relation, and furthermore, given it should be hard to find the corresponding . ## Some PreliminariesWe say that two numbers are Example
10 and 21 are relatively prime numbers. The prime factors of 10 are and . Thus, they have no prime factors in common. On the other hand and are not relatively prime. They have a prime factor in common. By convention, for all . Therefore, is taken to be relatively prime to every other non-zero number. ## Euler's TheoremYou can skip this section on a first read The answer comes from a result proved by Euler in 1741, called the
Totient of a Number
Given a number , its totient is the total number
of integers between and that are As an example, consider . The numbers that are relatively prime to include . Therefore . Interestingly, given a number , is rather hard to compute. But there are numbers for which it is relatively easy. Totient Function for Prime Numbers
Theorem: The totient function for a prime number is simply . The proof is simply to note that every number is relatively prime to . Another important result about Totient functions is as follows: Totient functions for products of primes
If and are prime numbers, such that , then . Let us look at all numbers between that have a common factor with . These can only be multiples of or multiples of . The reader can convince themselves that there are of these numbers. Therefore, . Here is what Euler's theorem says: Euler's Theorem
Let be relatively prime numbers. We have . The proof is rather elegant. Let be the set of all the numbers relatively prime with . Here and belongs to this set. Now consider . It is possible to show that and are, in fact, the same sets. Therefore, We therefore conclude that , or in other words . ## RSA SchemeWe are now ready to talk about the basic RSA scheme. The idea is to choose two different large prime numbers and compute . Let us assume , in general. We now wish to find a pair and for the public and private keys such that for any message , we have . Choosing any message between , we can use Totient's theorem to guarantee that . Therefore, if we choose any positive , we still obtain . Therefore, , for any . Our goal is now to find such that . In other words, . We follow the following steps to do so: Choose large primes with . Let . Let . Choose a number with that is relatively prime with . Compute such that for some . This is done using the *Euclidean Algorithm*.
Finally, we discard and simply retain . We publish as the public key and retain as the private key. Encryption
Encrypting a message is performed by . Likewise, decryption uses the private key . Decryption
Given cipher-text , we recover the original message as Let us illustrate this: Choose and . We have and . Let us choose . We have to find so that . We have and . Verify that . The private key is . The public key is .
Encryption: Take a message represented as a number from . The encrypted value of is . Example: Using public key and message , we have . To decrypt, we have to compute = . ## Breaking RSALet us assume that some one has access to the public key . What stops them from finding out , the secret key? After all, . Therefore, by factorizing , we can find and repeat the process for ourselves to compute and . Once is known then the whole scheme goes kaput.
In order to convince you that factoring a large number say digits is hard, your first programming assignment that will be out this monday asks you to try and write a factoring routine that given a number finds a prime factor of . You can use any method to do so. ## Combinatorially Hard ProblemsThere are problems in CS which do not have any known algorithms. The class of problems is called NP standing for Non-Deterministic Polynomial Time.
Naive Algorithm
int factor(int n){ int i; for (i = 0; i < n;++i) if (Divides(i,n)) return i; return NO_FACTOR; } Time taken to factor by best known algorithm is roughly . However, does that preclude a clever and faster algorithm? The best known factoring algorithm is the |