====== The Mathematical Guts of RSA Encryption ====== Quelle: https://www.cryer.co.uk/glossary/r/rsa/mathematical_guts_of_rsa_encryption.html The RSA algorithm was invented in 1978 by Ron Rivest, Adi Shamir, and Leonard Adleman. Here's the relatively easy to understand math behind RSA public key encryption. - Find //P// and //Q//, two large (e.g., 1024-bit) prime numbers. - Choose //E// such that //E// is greater than 1, //E// is less than //PQ//, and //E// and //(P-1)(Q-1)// are relatively prime, which means they have no prime factors in common. //E// does not have to be prime, but it must be odd. //(P-1)(Q-1)// can't be prime because it's an even number. - Compute //D// such that //(DE - 1)// is evenly divisible by //(P-1)(Q-1)//. Mathematicians write this as //DE = 1 (mod (P-1)(Q-1))//, and they call //D// the multiplicative inverse of //E//. This is easy to do -- simply find an integer //X// which causes //D = (X(P-1)(Q-1) + 1)/E// to be an integer, then use that value of //D//. - The encryption function is //C = (T^E) mod PQ//, where //C// is the ciphertext (a positive integer), //T// is the plaintext (a positive integer), and ^ indicates exponentiation. The message being encrypted, //T//, must be less than the modulus, //PQ//. - The decryption function is //T = (C^D) mod PQ//, where //C// is the ciphertext (a positive integer), //T// is the plaintext (a positive integer), and ^ indicates exponentiation. Your public key is the pair //(PQ, E)//. Your private key is the number //D// (reveal it to no one). The product //PQ// is the modulus (often called //N// in the literature). //E// is the public exponent. //D// is the secret exponent. You can publish your public key freely, because there are no known easy methods of calculating //D//, //P//, or //Q// given only //(PQ, E)// (your public key). //P// and //Q// need to be sufficiently long (>2048 bits), for this method to be considered secure. Here is an example of RSA encryption: P = 61 <- first prime number (destroy this after computing E and D) Q = 53 <- second prime number (destroy this after computing E and D) PQ = 3233 <- modulus (give this to others) E = 17 <- public exponent (give this to others) D = 2753 <- private exponent (keep this secret!) Your public key is (E,PQ). Your private key is D. The encryption function is: encrypt(T) = (T^E) mod PQ = (T^17) mod 3233 The decryption function is: decrypt(C) = (C^D) mod PQ = (C^2753) mod 3233 To encrypt the plaintext value 123, do this: encrypt(123) = (123^17) mod 3233 = 337587917446653715596592958817679803 mod 3233 = 855 To decrypt the ciphertext value 855, do this: decrypt(855) = (855^2753) mod 3233 = 123 One way to compute the value of 855^2753 mod 3233 is like this: 2753 = 101011000001 base 2, therefore 2753 = 1 + 2^6 + 2^7 + 2^9 + 2^11 = 1 + 64 + 128 + 512 + 2048 Consider this table of powers of 855: 855^1 = 855 (mod 3233) 855^2 = 367 (mod 3233) 855^4 = 367^2 (mod 3233) = 2136 (mod 3233) 855^8 = 2136^2 (mod 3233) = 733 (mod 3233) 855^16 = 733^2 (mod 3233) = 611 (mod 3233) 855^32 = 611^2 (mod 3233) = 1526 (mod 3233) 855^64 = 1526^2 (mod 3233) = 916 (mod 3233) 855^128 = 916^2 (mod 3233) = 1709 (mod 3233) 855^256 = 1709^2 (mod 3233) = 1282 (mod 3233) 855^512 = 1282^2 (mod 3233) = 1160 (mod 3233) 855^1024 = 1160^2 (mod 3233) = 672 (mod 3233) 855^2048 = 672^2 (mod 3233) = 2197 (mod 3233) Given the above, we know this: 855^2753 (mod 3233) = 855^(1 + 64 + 128 + 512 + 2048) (mod 3233) = 855^1 * 855^64 * 855^128 * 855^512 * 855^2048 (mod 3233) = 855 * 916 * 1709 * 1160 * 2197 (mod 3233) = 794 * 1709 * 1160 * 2197 (mod 3233) = 2319 * 1160 * 2197 (mod 3233) = 184 * 2197 (mod 3233) = 123 (mod 3233) = 123 If you have a computer program (such as the "bc" utility that comes with Linux), you can compute 855^2753 mod 3233 directly, like this: 855^2753 mod 3233 = 50432888958416068734422899127394466631453878360035509315554967564501 05562861208255997874424542811005438349865428933638493024645144150785 17209179665478263530709963803538732650089668607477182974582295034295 04079035818459409563779385865989368838083602840132509768620766977396 67533250542826093475735137988063256482639334453092594385562429233017 51977190016924916912809150596019178760171349725439279215696701789902 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305946326827861270203356980346143245697021484375 mod 3233 = 123