RSA Encryption Algorithm

I saw an Eclipse plug-in yesterday and want to register it. Decompilation. Locate the registration code. But I still cannot understand it. Biginteger is used. I also used code such as M. modpow (d, n. No way. I searched Google. The RSA algorithm is used.

Here is an introduction to the RSA algorithm on the Internet:

The RSA algorithm was specified ted in 1978 by Ron Rivest, Adi Shamir, and Leonard Adleman.

Here's the relatively easy to understand math behind RSA public key encryption.

1. FindPAndQ, Two large (e.g., 1024-bit) prime numbers.

2. ChooseESuch thatEIs greater than 1,EIs lessPQ, AndEAnd(P-1) (Q-1)AreRelatively prime, Which means they have no prime factors in common.EDoes not have to be prime, but it must be odd.(P-1) (Q-1)Can't be prime because it's an even number.

3. ComputeDSuch that(De-1)Is evenly divisible(P-1) (Q-1). Mathematicians write thisDe = 1 (mod (P-1) (Q-1 )), And they callDTheMultiplicative inverseOfE. This is easy to do -- simply find an integer x which causesD = (x (P-1) (Q-1) + 1)/ETo be an integer, then use that valueD.

4. The encryption function isC = (t ^ e) mod PQ, WhereCIs the ciphertext (a positive integer ),TIs the plaintext (a positive integer), and ^ indicates exponentiation. The message being encrypted,T, Must be less than the modulus,PQ.

5. The decryption function isT = (C ^ d) mod PQ, WhereCIs the ciphertext (a positive integer ),TIs the plaintext (a positive integer), and ^ indicates exponentiation.

YourPublic KeyIs the pair(PQ, E). YourPrivate KeyIs the numberD(Reveal it to no one). The productPQIsModulus(Often called N in the literature ).EIsPublic exponent.DIsSecret Exponent.

You can publish your public key freely, because there are no known easy methods of calculatingD,P, OrQGiven only(PQ, E)(Your public key). IfPAndQAre each 1024 bits long, the sun will burn out before the most powerful computers presently in existence can factor your modulusPAndQ.

An example of the RSA Algorithm

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 3233the decryption function is: decrypt (c) = (C ^ d) moD PQ = (C ^ 2753) mod 3233to encrypt the plaintext value 123, do this: encrypt (123) = (123 ^ 17) moD 3233 = 337587917446653715596592958817679803 mod 3233 = 855to decrypt the ciphertext value 855, do this: decrypt (855) = (855 ^ 2753) mod 3233 = 123one way to compute the value of 855 ^ 2753 mod 3233 is like this: 2753 = 101011000001 base 2, therefore2753 = 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 ^ 12 8*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) = 123if 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 = 5043288895841606873442289912739446663145387836003550931555 4967564501 05562861208255997874424542811005438349865428933638493024645144150785 17209179665478263530709963803538732650089668607477182974582295034295 04079035818459409563779385865989368838083602840132509768620766977396 67533250542826093475735137988063256482639334453092594385562429233017 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54234930424749053693992776114261796407100127643280428706083531594582 305946326827861270203356980346143245697021484375 mod 3233 = 123 find the publickey in the program and use rsatool to calculate the two factors p and q. However, my machine is too slow and I will give up after I come out.