Using this tedious (but simple for a computer) calculation, A can decode B’s message and obtain the original message N = 35. But since we only care about the result (mod 943), we can calculate all the partial results in that modulus, and by repeated squaring of 545, we can get all the exponents that are powers of 2. This looks like it will be a horrible calculation, and at first it seems like it is, but notice that 503 = 256+128+64+32+16+4+2+1 (this is just the binary expansion of 503). To find the decoding, A must calculate C d (mod N) = 545503(mod 943). To do so, he needs to find a number d such that ed = 1(mod (p − 1)(q − 1)), or in this case, such that 7d = 1(mod 880). The number 545 is the encoding that server sends to client. Server calculates the value of C = Me (mod N) = 357 (mod 943). Suppose, for this example, that the message is the number: M = 35. Now B knows enough to encode a message to Client. It is, however, not too hard to check to see if a number is prime-in other words to check to see that it cannot be factored. This means the computer would have to run for 10176 times the life of the universe to factor the large number. But for a 400 digit product, there are 10200 possibilities. Assuming a computer could test one million factorizations per second, in the lifetime of the universe, it could check 1024 possibilities. The lifetime of the universe is approximately 1018 seconds – an 18 digit number. Now it doesn’t take a computer long to try out 38000 possibilities, but what if the number to be factored is not ten digits, but rather 400 digits? The square-root of a number with 400 digits is a number with 200 digits. In this case, that square-root is roughly 38000.
For any size number, the computer has to check something that is of the order of the size of the square-root of the number to be factored. For example, it is easy to check that 31 and 37 multiply to 1147, but if you try to find the factor of 1147, then it will be horrible to calculate.Ī computer can factor that number fairly quickly, but (although there are some tricks) it basically does it by trying most of the possible combinations. It is based on the logic that it is easy to multiply large prime numbers, but factoring large numbers is very difficult. Cryptosystem is a way where a pair of keys work together to data encrypt and decrypt. Most importantly, RSA implements a public-key cryptosystem, as well as digital signatures or private key.
In 1978, Ron Rivest, Adi Shamir, and Leonard Adleman developed an algorithm as a cryptographic algorithm, which was essentially to replace the less secure National Bureau of Standards (NBS) algorithm. But Key Pair algorithm like RSA is one of the most secured and unbreakable Cryptosystems. For secure transport of data from one to another end, various encryption technologies have been introduced already.
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