## RSA Algorithm Explained: a step-by-step process

The art of information hiding, or Cryptography, is one of my favorite applications of Mathematics in computer science. Information Hiding dates from thousand of years ago in “non-standard hieroglyphs carved into monuments from Egypt‘s Old Kingdom (ca 4500+ years ago)” [History of Cryptography]. It has different real-world applications ranging from credit-card security to online transactions in protected websites (those with the security lock). There are different approaches of information hiding, and the one used in the applications cited is the one first described by Rivest, Shamir and Adleman, the RSA. A great post about public key cryptography is summarized by Dr. Duke O’Connor on his website post “RSA is 30, and counting” about the 30th anniversary of the paper gave the origin to RSA:

R. Rivest, A. Shamir, L. Adleman. A Method for Obtaining Digital Signatures and Public-Key Cryptosystems. Communications of the ACM, Vol. 21 (2), pp.120–126. 1978.

The application used on the Internet can be summarized in the following video:

If you want to teach your children :), take a look at this other video.

Dr. O’Connor summarizes two versions of the Mathematical foundation of the algorithm: one with the first flaw and the corrected version. The “fixed” version of the algorithm is what I had studied in school while studying Abstract Algebra, where the students were distributed into groups of three to implement their own Crypto systems and exchange keys and information in order to be tested. Of course, different assumptions were considered for the exercise. This post is just the output of the application I had developed in Java, what I called Cryptonline. I decided to use my free time to develop a Groovy on Grails Online Forum application where users can only see the messages if they login and use your public/private keys. The intention of the application is just for educational purposes. Read Dr. O’Connor’s post for background background information before moving on (in case you don’t have the background on it :D). As the T-Shirt says, it is just an algorithm.

Suppose you are transferring an important message over an unreliable communication channel such as a letter to someone, or simply to give your bank account to someone for transfer purposes.

As you can see, a reliable communication channel is the one where your message is encrypted and no one in between the sender and receiver can read it because the message is an arrangement that only you and the bank are capable of reading. The only way to do so is the creation of a keys that can lock and unlock the message hidden in the message. The RSA algorithm is based on the idea of Private and Public keys, where both are used to encode and decode the encrypted message. Suppose the string “John Smith” is the text to be decoded. Encryption is the process to hide the information on an Encrypted version and Decryption is the process of decoding the Encrypted text to its original format.

Of course, the idea is to have a key where you can distribute, so that only people you want to read the message can read it.

This is the basic idea of the algorithm. There are different variations of this algorithm depending on the application. Whenever you visit a website with the lock enabled means that any information exchanged between your computer and the visited website’s computer is encrypted.

By now you should have the background on this subject (considering you have gone to Dr. O’Connor’s website and read the Mathematical background). The source-code below is the actual output from Cryptonline, the one I developed back in 2000 at school. It is divided into 3 steps: Keys creation, Text Encryption and Text Decryption.

Public Key Creation

Let’s start from the Public Key.

-> Configuring random prime numbers P = 1277 Q = 2311 -> Calculating public keys N = P * Q; N = 2951147 -> FI = (P-1) * (Q-1); FI = 2947560 -> Calculating (E) While MCD(n >= 2 , 2947560) != 1 MCD(2 , 2947560) = 2 MCD(3 , 2947560) = 3 MCD(4 , 2947560) = 4 MCD(5 , 2947560) = 5 MCD(6 , 2947560) = 6 MCD(7 , 2947560) = 7 MCD(8 , 2947560) = 8 MCD(9 , 2947560) = 3 MCD(10 , 2947560) = 10 MCD(11 , 2947560) = 11 MCD(12 , 2947560) = 12 MCD(13 , 2947560) = 1 Correct! E = 13 Public Key (N,E) = (2951147 , 13)

Private Key Creation

The calculation of the Private Key is based on the matrix calculation of the Phi number, considering E.

-> Calculating private keys Initializing (p1,p2,p3) = (1, 0 , FI(n)) Initializing (q1,q2,q3) = (0, 1 , E )) While q3 != 0 quoc = p3 / q3 (t1,t2,t3) = (p1,p2,p3) - quoc * (q1,q2,q3) After, arrange the values: (p1,p2,p3) = (q1,q2,q3) (q1,q2,q3) = (t1,t2,t3) (13 <> 0) , then: quoc = 2947560 / 13 = 226735 (t1,t2,t3) = (0,1,13) - 226735 * (1,-226735,5) = (1,-226735,5) (p1,p2,p3) = (1,-226735,5) (q1,q2,q3) = (1,-226735,5) (5 <> 0) , then: quoc = 13 / 5 = 2 (t1,t2,t3) = (1,-226735,5) - 2 * (-2,453471,3) = (-2,453471,3) (p1,p2,p3) = (-2,453471,3) (q1,q2,q3) = (-2,453471,3) (3 <> 0) , then: quoc = 5 / 3 = 1 (t1,t2,t3) = (-2,453471,3) - 1 * (3,-680206,2) = (3,-680206,2) (p1,p2,p3) = (3,-680206,2) (q1,q2,q3) = (3,-680206,2) (2 <> 0) , then: quoc = 3 / 2 = 1 (t1,t2,t3) = (3,-680206,2) - 1 * (-5,1133677,1) = (-5,1133677,1) (p1,p2,p3) = (-5,1133677,1) (q1,q2,q3) = (-5,1133677,1) (1 <> 0) , then: quoc = 2 / 1 = 2 (t1,t2,t3) = (-5,1133677,1) - 2 * (13,-2947560,0) = (13,-2947560,0) (p1,p2,p3) = (13,-2947560,0) (q1,q2,q3) = (13,-2947560,0) q3 is zero(0). Now, verify the value of p2. In case of negative, invert it by summing it with FI. (represent the negative number of z(n) by a positive.) u2 = 1133677; D = u2; D = 1133677 Private Key (N,D) = (2951147, 1133677);

Using the Private and Public Keys

To summarize, the program outputs the keys used throughout the application.

#### All RSA Information #### Public Key (N, E) = (2951147, 13) Private Key (N, D) = (2951147, 1133677)

Of course you cannot give the original Prime Numbers to anybody since they are the factors that created the public and private key. So, the only key you can give away to other people is the Public one. Now, in order to illustrate Encryption and Decryption, my encryption machine uses ASCII-based character mapping for the Mathematical calculations. Consider my blog’s title as the input “Marcello de Sales: because solving problems is addicting”. Each character of the string is transformed into the ASCII added other numbers.

Encryption Process

-> Original Message Marcello de Sales: because solving problems is addicting -> Setting the receiver's public key (N , E) = (2951147 , 13) -> Transforming the message to ASCII code 177197214199201208208211132200201132183197208201215158132198201199197217215201132215211208218205210203132212214211198208201209215132205215132197200200205199216205210203 -> Configuring randomly selected blocks from the ASCII message Bloco(x) = x ^ E mod N Block(17) = 17 ^ 13 mod 2951147 = 2920887 Block(71) = 71 ^ 13 mod 2951147 = 1483408 Block(972) = 972 ^ 13 mod 2951147 = 363316 Block(1419) = 1419 ^ 13 mod 2951147 = 1419505 Block(920) = 920 ^ 13 mod 2951147 = 213548 Block(1) = 1 ^ 13 mod 2951147 = 1 Block(20) = 20 ^ 13 mod 2951147 = 93651 Block(8) = 8 ^ 13 mod 2951147 = 1394993 Block(2082) = 2082 ^ 13 mod 2951147 = 2878680 Block(1113) = 1113 ^ 13 mod 2951147 = 770001 Block(2200) = 2200 ^ 13 mod 2951147 = 2301917 Block(20113) = 20113 ^ 13 mod 2951147 = 787047 Block(2183) = 2183 ^ 13 mod 2951147 = 424239 Block(19) = 19 ^ 13 mod 2951147 = 1557862 Block(7) = 7 ^ 13 mod 2951147 = 2854397 Block(208) = 208 ^ 13 mod 2951147 = 375871 Block(2012) = 2012 ^ 13 mod 2951147 = 491468 Block(151) = 151 ^ 13 mod 2951147 = 2348470 Block(58) = 58 ^ 13 mod 2951147 = 966721 Block(13219) = 13219 ^ 13 mod 2951147 = 2596853 Block(820) = 820 ^ 13 mod 2951147 = 1336058 Block(11991) = 11991 ^ 13 mod 2951147 = 2624815 Block(97) = 97 ^ 13 mod 2951147 = 1340264 Block(21721) = 21721 ^ 13 mod 2951147 = 1760166 Block(52011) = 52011 ^ 13 mod 2951147 = 1685895 Block(32215) = 32215 ^ 13 mod 2951147 = 1202590 Block(21) = 21 ^ 13 mod 2951147 = 2752293 Block(1208) = 1208 ^ 13 mod 2951147 = 1414540 Block(21820) = 21820 ^ 13 mod 2951147 = 1733373 Block(5) = 5 ^ 13 mod 2951147 = 1879414 Block(21020) = 21020 ^ 13 mod 2951147 = 310870 Block(3132) = 3132 ^ 13 mod 2951147 = 519822 Block(212) = 212 ^ 13 mod 2951147 = 1315135 Block(2142) = 2142 ^ 13 mod 2951147 = 2430603 Block(1119) = 1119 ^ 13 mod 2951147 = 748920 Block(8208) = 8208 ^ 13 mod 2951147 = 2808982 Block(20) = 20 ^ 13 mod 2951147 = 93651 Block(1209) = 1209 ^ 13 mod 2951147 = 906866 Block(215) = 215 ^ 13 mod 2951147 = 396673 Block(13) = 13 ^ 13 mod 2951147 = 2564672 Block(2205) = 2205 ^ 13 mod 2951147 = 337248 Block(2) = 2 ^ 13 mod 2951147 = 8192 Block(151) = 151 ^ 13 mod 2951147 = 2348470 Block(32) = 32 ^ 13 mod 2951147 = 1191513 Block(197200) = 197200 ^ 13 mod 2951147 = 2266852 Block(200) = 200 ^ 13 mod 2951147 = 104075 Block(20) = 20 ^ 13 mod 2951147 = 93651 Block(519) = 519 ^ 13 mod 2951147 = 1459225 Block(9) = 9 ^ 13 mod 2951147 = 1601171 Block(21620) = 21620 ^ 13 mod 2951147 = 2477239 Block(5210) = 5210 ^ 13 mod 2951147 = 1598948 Block(203) = 203 ^ 13 mod 2951147 = 644537 -> Encrypted Message 2920887-1483408-363316-1419505-213548-1-93651-1394993-2878680-770001-2301917-787047-424239-1557862-2854397-375871-491468-2348470-966721-2596853-1336058-2624815-1340264-1760166-1685895-1202590-2752293-1414540-1733373-1879414-310870-519822-1315135-2430603-748920-2808982-93651-906866-396673-2564672-337248-8192-2348470-1191513-2266852-104075-93651-1459225-1601171-2477239-1598948-644537

Decryption Process

Upon receiving the encrypted message, the receiver needs to use the public key from the sender in order to decrypt the message. The receiver has the same encryption machine, but needs your public key in order to decipher it. The program developed decrypts the message as shown below.

-> Encrypted Message 2920887-1483408-363316-1419505-213548-1-93651-1394993-2878680-770001-2301917-787047-424239-1557862-2854397-375871-491468-2348470-966721-2596853-1336058-2624815-1340264-1760166-1685895-1202590-2752293-1414540-1733373-1879414-310870-519822-1315135-2430603-748920-2808982-93651-906866-396673-2564672-337248-8192-2348470-1191513-2266852-104075-93651-1459225-1601171-2477239-1598948-644537 -> Setting the private key (N , D) = (2951147 , 1133677) -> Decripting each block Ascii(x) = x ^ D mod N Ascii(2920887) = 2920887 ^ 1133677 mod 2951147 = 17 Ascii(1483408) = 1483408 ^ 1133677 mod 2951147 = 71 Ascii(363316) = 363316 ^ 1133677 mod 2951147 = 972 Ascii(1419505) = 1419505 ^ 1133677 mod 2951147 = 1419 Ascii(213548) = 213548 ^ 1133677 mod 2951147 = 920 Ascii(1) = 1 ^ 1133677 mod 2951147 = 1 Ascii(93651) = 93651 ^ 1133677 mod 2951147 = 20 Ascii(1394993) = 1394993 ^ 1133677 mod 2951147 = 8 Ascii(2878680) = 2878680 ^ 1133677 mod 2951147 = 2082 Ascii(770001) = 770001 ^ 1133677 mod 2951147 = 1113 Ascii(2301917) = 2301917 ^ 1133677 mod 2951147 = 2200 Ascii(787047) = 787047 ^ 1133677 mod 2951147 = 20113 Ascii(424239) = 424239 ^ 1133677 mod 2951147 = 2183 Ascii(1557862) = 1557862 ^ 1133677 mod 2951147 = 19 Ascii(2854397) = 2854397 ^ 1133677 mod 2951147 = 7 Ascii(375871) = 375871 ^ 1133677 mod 2951147 = 208 Ascii(491468) = 491468 ^ 1133677 mod 2951147 = 2012 Ascii(2348470) = 2348470 ^ 1133677 mod 2951147 = 151 Ascii(966721) = 966721 ^ 1133677 mod 2951147 = 58 Ascii(2596853) = 2596853 ^ 1133677 mod 2951147 = 13219 Ascii(1336058) = 1336058 ^ 1133677 mod 2951147 = 820 Ascii(2624815) = 2624815 ^ 1133677 mod 2951147 = 11991 Ascii(1340264) = 1340264 ^ 1133677 mod 2951147 = 97 Ascii(1760166) = 1760166 ^ 1133677 mod 2951147 = 21721 Ascii(1685895) = 1685895 ^ 1133677 mod 2951147 = 52011 Ascii(1202590) = 1202590 ^ 1133677 mod 2951147 = 32215 Ascii(2752293) = 2752293 ^ 1133677 mod 2951147 = 21 Ascii(1414540) = 1414540 ^ 1133677 mod 2951147 = 1208 Ascii(1733373) = 1733373 ^ 1133677 mod 2951147 = 21820 Ascii(1879414) = 1879414 ^ 1133677 mod 2951147 = 5 Ascii(310870) = 310870 ^ 1133677 mod 2951147 = 21020 Ascii(519822) = 519822 ^ 1133677 mod 2951147 = 3132 Ascii(1315135) = 1315135 ^ 1133677 mod 2951147 = 212 Ascii(2430603) = 2430603 ^ 1133677 mod 2951147 = 2142 Ascii(748920) = 748920 ^ 1133677 mod 2951147 = 1119 Ascii(2808982) = 2808982 ^ 1133677 mod 2951147 = 8208 Ascii(93651) = 93651 ^ 1133677 mod 2951147 = 20 Ascii(906866) = 906866 ^ 1133677 mod 2951147 = 1209 Ascii(396673) = 396673 ^ 1133677 mod 2951147 = 215 Ascii(2564672) = 2564672 ^ 1133677 mod 2951147 = 13 Ascii(337248) = 337248 ^ 1133677 mod 2951147 = 2205 Ascii(8192) = 8192 ^ 1133677 mod 2951147 = 2 Ascii(2348470) = 2348470 ^ 1133677 mod 2951147 = 151 Ascii(1191513) = 1191513 ^ 1133677 mod 2951147 = 32 Ascii(2266852) = 2266852 ^ 1133677 mod 2951147 = 197200 Ascii(104075) = 104075 ^ 1133677 mod 2951147 = 200 Ascii(93651) = 93651 ^ 1133677 mod 2951147 = 20 Ascii(1459225) = 1459225 ^ 1133677 mod 2951147 = 519 Ascii(1601171) = 1601171 ^ 1133677 mod 2951147 = 9 Ascii(2477239) = 2477239 ^ 1133677 mod 2951147 = 21620 Ascii(1598948) = 1598948 ^ 1133677 mod 2951147 = 5210 Ascii(644537) = 644537 ^ 1133677 mod 2951147 = 203 -> Complete message in ASCII 177197214199201208208211132200201132183197208201215158132198201199197217215201132215211208218205210203132212214211198208201209215132205215132197200200205199216205210203 -> Original Message Marcello de Sales: because solving problems is addicting

This is fun! Now you can talk anything with your peers :D. Different practical applications are in Internet chat rooms. I’ve used Adium for Mac in order to chat through a secure communication channel over the Internet :). I keep thinking which the keys are when I’m communicating (when I’m bored…).

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