ࡱ> jli` Pbjbj .`-84`vj* %"&&&'''suuuuuuuuuuuuu$wh2zXu.}''..u&&gvQQQ. 8&&suQ.suQQRr3u& DkE<ssu0v0`vsbztDz<3uz3u@')Q+4O,{'''uu8K'''`v.... Chris Farley Research Paper RSA Public-key Cryptosystem Suppose Alice and Bob want to communicate with each other, but they do not want others to be able to ease drop on their conversations. For example, Alice and Bob may be working on top-secret research or they may be allies in a war. They would need to develop their own system for communication like their own language or alphabet. However, this would not necessarily ensure that they could protect the translation from an attacker. They need a way of converting information and then decoding the converted information so that it is only understood by each other. The process by which they will achieve this is cryptography, also known as cryptology. The particular scheme that they employ is a cryptosystem. Cryptosystems are made up of three basic parts: the encryption algorithm, the decryption algorithm, and the key(s). The encryption algorithm is the algorithm used to encode an original, or a plaintext message. The decryption algorithm is the reverse process of the encryption algorithm. With the decryption algorithm, the user converts the encoded message back to its original plaintext message. The key system is used during the process of encrypting and decrypting messages. Generally, an encryption key is used to encrypt messages, whereas a decryption key is used decrypt messages. The RSA algorithm is used in cryptography as a public-key cryptosystem. This algorithm was the first known to be suitable for signing as well as encryption. Because of this, it was one of the first great advancements in public-key cryptology. RSA is still widely used and is believed to be secure given sufficiently long keys. This is because RSA is based on the difficulty of factoring large prime numbers. Ron Rivest, Adi Shamir, and Len Adleman described the RSA algorithm at MIT in 1977. The algorithms name, RSA, was derived from the initials of their last names. The algorithm was patented by MIT in the United States in 1983. It was given the U.S. Patent number 4,405,829, expiring on September 21, 2000 (Wikipedia, RSA). A British mathematician, Clifford Cocks, supposedly had written an equivalent system to RSA in 1973. However, the research was not pursued due to the lack of and the cost of computers that were needed to execute the algorithm. This discovery was not revealed until 1997 because of the top-secret classification given by GCHQ (UK Government Communications Head Quarters), of which, Cocks was affiliated. Now that we have a little background about RSA, it is time to answer how Alice can securely communicate with Bob. How do we go about generating the keys that we need to securely send the message? If Bob wants to have Alice send him a secure message over an insecure medium, then he needs to take the following steps to generate the public and private keys: Choose two large primes, p and q, randomly and independently of each other; however, be sure that they are very large so that their product will be of sufficient size. Also, the primes are to be kept secret. Compute the product N = pq. The product, N, is to be made public and is known as the modulus. Compute the totient  EMBED Equation.3  The totient function is Eulers phi function (see notes section). Choose an integer e (which is to be made public), where  EMBED Equation.3  such that  EMBED Equation.3 or equivalently, e is coprime or relatively prime to EMBED Equation.3  In general, if EMBED Equation.3 then a and b are relatively prime, and  EMBED Equation.3 is a congruence. Note that this can be found using the Euclidean Algorithm. Compute the private key, d, which is the multiplicative inverse of  EMBED Equation.3 i.e., find an integer d with  EMBED Equation.3  In general, solve for d in the equation  EMBED Equation.3  The existence of d follows from the fact that if given two integers, a and b, where the  EMBED Equation.3 then b is invertible (mod a) and b has a multiplicative inverse in EMBED Equation.3 . This can also be found with the Euclidean Algorithm. That is how we generate our keys. The public key consists of the modulus, N, and the public exponent, also referred to as the encryption exponent, e. The private key consists of, again, the public modulus, N, and the private exponent, d, which is also called the decryption exponent. The value of d must be kept secret, as well as the values of p, q, and EMBED Equation.3  Now that we have our keys, we can start sending messages. How do we use our keys to send encrypted messages and then decrypt them? To explain, we will return to our hypothetical guinea pigs, Alice and Bob. Suppose Alice wishes to send a message M to Bob. Alice converts the message M to an integer, m < N, by using whatever method she chooses (I will briefly discuss different types of methods that are generally used to do this in other sections of the paper). Note that if  EMBED Equation.3 then Alice would need to break it up into groups, where each group is a number  EMBED Equation.3 and the encryption procedure would be done on all groups. However, for this example, we will assume that m < N. Alice has m, and given the public key N and e, she then converts m to the ciphertext, y, by using the following equation:  EMBED Equation.3  This can be done quickly using the method of exponentiation by squaring (see notes section), but is usually done with a computer. After Alice converts the message to cyphertext, she sends it to Bob. When Bob receives the message, he uses his private key, d, to recover m from y by doing the inverse modular exponentiation process:  EMBED Equation.3  That was easy enough, right? However, you may be asking how we can be sure that when Bob receives the ciphered message, and then deciphers it, that he will arrive at the original message sent by Alice? In other words, we need to verify that  EMBED Equation.3 (this notation is described in the notes section, briefly; we say  EMBED Equation.3 is congruent to m modulo N, where N is the modulus of the congruence, also written  EMBED Equation.3 , Michon, G. P.) For a quick explanation, the decryption procedure works because d mod N is the multiplicative inverse of e. Thus,  EMBED Equation.3 is the inverse of  EMBED Equation.3  To elaborate,  EMBED Equation.3 also,  EMBED Equation.3 and  EMBED Equation.3  Using Fermats little theorem (see notes section) we conclude that  EMBED Equation.3 and EMBED Equation.3  Given that p and q are distinct primes, the Chinese remainder theorem (see notes section) says that these two congruences can be rewritten as EMBED Equation.3 , and finally,  EMBED Equation.3 % (Wikipedia, RSA) In order to exemplify the RSA encryption/decryption process with a working example, we will use very small numbers that are not recommended to be used for real RSA processes. Suppose Alice wants to send the message Hello Bob to Bob. Also, assume that Alice and Bob have decided to convert letters to numbers by using each letters corresponding two-digit representational position in the alphabet. So our number m is to represent the message Hello Bob. Before we go further, we need to generate our keys. First, we will pick our two primes which are to be kept secret. Since we are using relatively small numbers, we let p = 53 and q = 41. Next, we compute N = pq = EMBED Equation.3  which is the modulus, a part of the public key. Then, we choose e such that it is coprime to EMBED Equation.3  For this example, we will choose e = 7, verified by  EMBED Equation.3  Remember that e, the encryption exponent, is also part of the public key. From here, we can find d by solving for it in the equation 1 = 7d mod 2080. In this case, d = 1783 will work because  EMBED Equation.3 (CITE) Now that we have generated our public key, (N, e), and our private key, (N, d), we are ready to proceed. We will assume that Bob made the public key and sent it to Alice so that she could transfer encrypted messages to him. We will also assume that Bob has our private key, which he will use to decrypt the message. Alice transforms Hello Bob to m = 080512121500021502 (for our example any blank space is represented by 00). Since the number m is larger than the modulus N, Alice must split it up. To do so, she splits m into six different numbers, each containing three digits, ensuring that each EMBED Equation.3 :  EMBED Equation.3  Next, for each EMBED Equation.3 , Alice will convert the cyphertext y, where y is also made up of six numbers corresponding to each of the  EMBED Equation.3  To do this, Alice computes EMBED Equation.3 :  EMBED Equation.3  Alice then transmits all six of these messages to Bob. When Bob gets around to checking his messages, he realizes that Alice has sent him one. Excited, he quickly applies his private key, d = 1783 to decipher the cyphertext. To do so, he computes  EMBED Equation.3  for each EMBED Equation.3 :  EMBED Equation.3  I will explain how one can compute these large modular exponentiations later (see notes section), for now, I will continue with the decrypting process. Bob now has all of the pieces that he needs to decipher the message. Bob groups the pieces together and gets the number 080512121500021502. He then splits the number up again by grouping in pairs: 08 05 12 12 15 00 02 12 02. Since Bob knows that the numbers correspond to positions of letters in the alphabet, he can convert the message one step further, deciphering the message to discover that Alice says, Hello Bob. As mentioned earlier, the security of relies on the fact it is difficult to factor sufficiently large numbers. No polynomial-time method for factoring large integers on a classical computer has been found yet. However, it has not been proven that none exists. As of November 8, 2005, the largest number factored by general purpose methods was 640 bits long (known as the RSA-640 number) which was a 193 digit number. This factorization was done on 80 CPUs working together and took approximately four and a half months, total. The prize that the German BSI team received for factoring this number was $20,000. Because of the different methods of factorization that are being used with the help of linked computers or super-computers, the current recommendation for the product of primes, N, is for it to be is at least 2048 bits long. However, if quantum computing reaches a sufficient level, as it is predicted to be no earlier than 2015, it potentially can perform factorization in polynomial time, rendering RSA and related algorithms obsolete. How do we know that the message we receive is the sent by our correspondent? For this, RSA provides for message signing which entails the sender producing a hash value of the message to be sent, raising that value to the power of d mod N (as done when decrypting), and attaching it as a signature to the message. When the receiver gets the signed message, he raises the signature to the power of e mod N (as done when encrypting), and compares the resulting hash value with the messages actual hash value, making sure they agree, therefore verifying the signature. (Wikipedia, RSA) There are various different methods that are used to pad numbers to messages to enhance the security of the messages. These methods are commonly referred to as padding schemes. Although, we will not fully discuss these schemes, I think that it is pertinent to know the value of applying these techniques for a practical application. For example, if a sender did not use padding schemes, then the security of their message may be flawed by the following things: Both of the values of m = 0 or m = 1 will produce the same ciphertext due to the properties of exponentiation. If the encryption is done with small encryption exponents and small message values, then the result of  EMBED Equation.3 could be strictly less than the value of the modulus n. If this is the case, then the ciphertext is easily discovered by taking the eth root of it (without using the modulus). Because RSA uses public keys, then an attacker could develop a ciphertext library of resultant ciphers of encryptions using the public key. If the ciphertext that are intercepted match any of the librarys ciphers, then the attacker would know the resultant plaintext. Again, there are many different techniques that are employed to securely pad plaintext. This paper has some references which describe some of these schemes in further detail. The computation of RSA renders it to be slower than other cryptosystems. Because of this, typical applications of RSA would involve sending credit card numbers, very short messages, map coordinates, or other relatively small numbers. If someone wished to send a large message, they usually would encrypt the message with a different cryptography method such as DES. Then they would encrypt a key to decrypt the message with RSA. Of course, the key would be relatively small. After which, the sender would transmit both the RSA encrypted key to the message and the encrypted message itself. There are lots of other security issues and general considerations regarding RSA. For more information, consult the references listed at the end of this paper. For now, we will elaborate on the mathematical backbone of RSA. Notes Section: How It All Works This section presents some principles of number theory and definitions that are used throughout this paper. First, for congruences, we have several forms of notation. For example, a is congruent to b modulo N is a congruence where N is the modulus of the congruence. Mathematically, this congruence can be expressed with the following notations, of which, the first two are used in this paper:  EMBED Equation.3  Next, these ideas are the foundational tools that make the RSA cryptosystem work. Prime numbers are those that are greater than one, which only has factors of one and itself. Two numbers are said to be relatively prime or coprime if the two numbers have no common factors other than one. The largest integer that divides two integers, say, a and b, is known as the greatest common divisor of a and b, denoted  EMBED Equation.3  If two numbers, a and b, are coprime, then  EMBED Equation.3 . The converse is also true. A corollary to this idea is that there exist integers x and y such that  EMBED Equation.3  (Bzouts identity). The Euler phi function or totient function, denoted  EMBED Equation.3  is defined as the number of integers from 1 to N that are coprime to N, where such numbers are totatives of N. Also, for any prime p,  EMBED Equation.3  When we say N divides a, we use the notation N | a. Also, there exists an integer k such that a = Nk. The common notation,  EMBED Equation.3 means a and b have the same remainder when divided by N, or N | a-b. If  EMBED Equation.3 and  EMBED Equation.3 then  EMBED Equation.3 and  EMBED Equation.3  If given that p and q are coprime, and p | a and q | a, then pq | a. Also, if pq | a then p | a and q | a, for all p and q. If p and q are coprime, then EMBED Equation.3  The following section describes the theorems that RSA cryptography is based on: Taken from the Chinese Remainder Theorem: Let p and q be integers, not necessarily prime, such that they are coprime. If  EMBED Equation.3 and  EMBED Equation.3  then we have  EMBED Equation.3  Proof: Since  EMBED Equation.3  then p | a-b. Also, since  EMBED Equation.3  then q | a-b. But p and q are coprime, thus pq | a-b. Therefore,  EMBED Equation.3 % (Davis, T.) Fermat s Little Theorem: For any integer a and any prime number p,  EMBED Equation.3  If a and p are coprime, then  EMBED Equation.3  or, equivalently, EMBED Equation.3  This Theorem has many proofs, for ours, we will prove it by using group theory. For this we will need to recognize that  EMBED Equation.3 forms a group under the operation of multiplication modulo p and also, we will need to use Lagranges Theorem: The order of any subgroup divides the order of the whole group. Proof: Let  EMBED Equation.3 form a group under the operation of multiplication modulo p. Choose  EMBED Equation.3  Let k be the order of a, so that  EMBED Equation.3  By Lagranges theorem, k divides the order of G, which is  EMBED Equation.3  So,  EMBED Equation.3  for some positive integer m. Thus,  EMBED Equation.3 % (Wikipedia, Proofs of Fermat s Little Theorem) Fermat-Euler Theorem or Euler s Generalization of Fermat s Little Theorem: For any number a coprime to n, we have  EMBED Equation.3 where  EMBED Equation.3  is Eulers phi function (as previously described). Since this is a generalization of Fermats Little Theorem, we will not display a formal proof. From here, we have all of the tools needed to prove the main result of the RSA process. The following is a proof that will use the abovementioned number theory and theorems to prove the main result of the RSA scheme. Choose two large primes, p and q, randomly and independently of each other. Let  EMBED Equation.3  also, define the totient function  EMBED Equation.3  Next, let e be an integer such that  EMBED Equation.3  (i.e. e is coprime to EMBED Equation.3 ). Then, let d be a number such that  EMBED Equation.3  (A further elaboration of how to generate d is given following our proof). Given the message m, an integer such that  EMBED Equation.3  define the encoded message, y, as EMBED Equation.3  We need to show that the decoded message is given by  EMBED Equation.3  Proof of main result: Since we are given that  EMBED Equation.3 we can express this congruence as  EMBED Equation.3 that is  EMBED Equation.3  divides  EMBED Equation.3  Since p and q are coprime, we have  EMBED Equation.3  Therefore,  EMBED Equation.3  and we can also express this as  EMBED Equation.3  and  EMBED Equation.3  Since  EMBED Equation.3  then  EMBED Equation.3  an integer k such that  EMBED Equation.3  or, because p is prime,  EMBED Equation.3  Now, looking at the general case of  EMBED Equation.3  We can form the congruence  EMBED Equation.3  or equivalently,  EMBED Equation.3  As shown above, we can substitute for  EMBED Equation.3  and we have,  EMBED Equation.3  Since p is prime, then our integer, a, will be either coprime to p, or it will be a multiple of p (i.e.  EMBED Equation.3  Case 1: In the case where a is coprime to p, from Fermats Little Theorem, we have  EMBED Equation.3  We can raise this to the power of an integer k getting  EMBED Equation.3  Thus, our congruence can be rewritten as  EMBED Equation.3  Case 2: Considering the other case where a is a multiple of p, we recognize that if  EMBED Equation.3 then  EMBED Equation.3  From this fact, we have  EMBED Equation.3  But, since  EMBED Equation.3  we have  EMBED Equation.3  Therefore, because  EMBED Equation.3  the congruence  EMBED Equation.3  still holds true. By the same argument, we can show that because  EMBED Equation.3  we have  EMBED Equation.3  We have shown that  EMBED Equation.3 and  EMBED Equation.3  Thus, because p and q are coprime, by the Chinese Remainder Theorem,  EMBED Equation.3  We can now relate this to our original equation, as given,  EMBED Equation.3  Now we have  EMBED Equation.3  But, as a result shown above,  EMBED Equation.3  Hence,  EMBED Equation.3 and  EMBED Equation.3  At this point, it is important to mention that expressed as a congruence, this relationship does not yield a one-to-one correspondence between our message m and our ciphertext y. But, because the calculation is limited to EMBED Equation.3 , we are limiting the value to be less than our modulus, N. Therefore, there is only one integer m, satisfying  EMBED Equation.3  Thus,  EMBED Equation.3  % (DI Management) Further elaboration on the generation of d, with terms defined as above, is as follows: Given the integer e, we will need to find d such that  EMBED Equation.3  To find this d, we will use Euler s Generalization of Fermat s Little Theorem. By this theorem we get  EMBED Equation.3  so  EMBED Equation.3  is a suitable value for d (Davis, T.). The following section describes, as promised, the computation of large modular exponentiations: Look at the following computation, as seen previously.  EMBED Equation.3  Although this calculation looks tough, it can be done by taking the binary expansion of the exponent, d, in order to calculate partial results modulo N, otherwise known as the method of modular exponentiation by squaring. To elaborate, consider our d = 1783. This can also be written as d =1024 + 512 + 128 + 64+ 32 + 16 + 4 + 2 + 1 = 1783. Therefore, if we want to calculate EMBED Equation.3 , we do the following:  EMBED Equation.3  Now we need to calculate the partial results modulo N, and following the repeated squaring pattern of 200, we will get all the exponents that have a power of 2. As we obtain these, we will exclude the ones that are not used in the binary expansion of 1783 (we will not need 256 or 8):  EMBED Equation.3  The result we will generate from this is:  EMBED Equation.3  That is exactly what we were looking for. In this case, the computation of  EMBED Equation.3  is also too challenging for your calculator, but you can repeat the preceding process on it and then use its result to finish the computation (Davis, T.). Random quote: Mathematics is the Queen of sciences, and arithmetic the Queen of mathematics. Carl Friedrich Gauss (1777-1855). Reference: Davis, T., RSA Encryption. Geometer, Math Circles. Retrieved on November 11, 2005 from the World Wide Web:  HYPERLINK "http://www.geometer.org/mathcircles" http://www.geometer.org/mathcircles DI Management, RSA Theory. DI Management Services, 2000-2004. Retrieved on December 4, 2005 from the World Wide Web:  HYPERLINK "http://www.di-mgt.com.au/rsa_theory.html" http://www.di-mgt.com.au/rsa_theory.html DI Management, RSA Algorithm. DI Management Services, 2000-2004. Retrieved on December 4, 2005 from the World Wide Web:  HYPERLINK "http://www.di-mgt.com.au/rsa_theory.html" http://www.di-mgt.com.au/rsa_theory.html Gilbert, J., Gilbert, L., Elements of Modern Algebra. Brooks/Cole, Thompson Learning, 2000. ISBN 0-534-37351-8. Gottesman, D., Lo Hoi-Kwong, Figure 2 from From Quantum Cheating to Quantum Security. Physics Today Online, 2001 American Institute of Physics. Retrieved on November 8, 2005 from the World Wide Web:  HYPERLINK "http://www.physicstoday.org/pt/vol-53/iss-11/captions/p22cap2.html" http://www.physicstoday.org/pt/vol-53/iss-11/captions/p22cap2.html Johnsonbaugh, R., Discrete Mathematics. Prentice Hall 2001. ISBN 0-13-089008-1. Lewand, R. E., Cryptological Mathematics. The Mathematical Association of America Inc., 2000. Library of Congress Catalog Card Number 00-105256. ISBN 0-88385-719-7. Michon, G. P., Final Answers: Modular Arithmetic. 2000-2005 Gerald P. 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FMicrosoft Equation 3.0 DS Equation Equation.39q$bhV  ey  ei  e=m  ei  eemodNOle CompObjfObjInfoEquation Native ~_1195329048F@ DkE@ DkEOle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q$ p_$  ey  e1  e=080  e7mod2173=200  y  e4  e=500  e7mod2173=1355  ey  e2 Equation Native %_1195329078F@ DkE@ DkEOle CompObjf e=512  e7mod2173=1730  y  e5  e=021  e7mod2173=1837  ey  e3  e=121  e7mod2173=1102  y  e6  e=502  e7mod2173=592 FMicrosoft Equation 3.0 DS Equation Equation.39q$fx;  em  ei  e=y  ei  edmodN,ObjInfoEquation Native _1195316908RF0TDkE0TDkEOle CompObjfObjInfoEquation Native ?_1195329086F0TDkE0TDkE FMicrosoft Equation 3.0 DS Equation Equation.39q$#>  ey  ei  FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native m     !$'*-058;>ADGJMPSVY\_behknqtwz}$Q0  em  e1  e=200  e1783mod2173=080  m  e4  e=1355  e1783mod2173=500  em  e2  e=1730  e1783mod2173=512  m  e5  e=1837  e1783mod2173=021  em  e3  e=1102  e1783mod2173=121  m  e6  e=592  e1783mod2173=502 FMicrosoft Equation 3.0 DS Eq_1196011641F0TDkE0TDkEOle CompObjfObjInfouation Equation.39q$"k  em  ee FMicrosoft Equation 3.0 DS Equation Equation.39qpTb  eaa"bEquation Native >_1195279216gF0TDkE0TDkEOle CompObjfObjInfoEquation Native _1195226666F0TDkE0TDkEOle (modN)  eamodN=b  eaa"b[N]  e[a]10=[b]10 FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjfObjInfoEquation Native  M_1195227406F0TDkE0TDkE1pTb gcd(a,b). 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