While I can't read kanji, I'll forever remember Algebra and its application on the RSA cryptography. Good thing Flandre is actually a centuries old vampire, because this is not for kids :P
*sigh* Fujii Tatsuya, I hate you. *sees book-eating Flan* Fuji Tatsuya, I love you.
EDIT: Done. Please check it over; it's been a while since I did something with RSA.
p=7, q=13, n=91, phi(n)=72.
We have x^11 mod 91 = 5.
Since phi(n)=72, we'll find the private key of 11 by computing a in 11a+72b=1, and we have a=59. Then we have x mod 91 = 5^59, and calculate x mod 91 = 73.
Wow. Even with small primes like 7 and 13, we ended up having to do a lot of calculation. If you have a PC, it can be pretty fun using the calculator functions X^Y and MOD. (Choose the Scientific Calculator option from the calculator's menu.)Because you have a scientific calculator on your PC, manual calculations aren't necessary. Try decrypting the ciphertext on the next page.RSA Cryptography[What is RSA Cryptography?]
RSA Cryptography is an example of a Public Key Cryptographical Algorithm, where two extremely large prime numbers p and q are used. In order for the sender and receiver to communicate with each other, the sender calculates pq = n.
We have a message x, and phi(n)=(p-1)(q-1). We choose public key e such that e is coprime with phi(n), and compute ciphertext x^e. The only information available to the sender is the message x, public key e, and n.
The receiver has a private key d such that ed mod phi(n) = 1, and raise the ciphertext to its dth power (according to Euler's Theorem, x^(ed) mod n = x) to obtain the original plaintext.
Since phi(n) can be obtained easily by factoring n through heuristics, in practice extremely large numbers of p and q (2048-bit long) are used to prevent the factorizaion from being completed in a practical timeframe. This is how RSA stays secure.
Let's see an example with some really small prime numbers.Column [RSA Cryptography]
Today I'll be the instructor. Number theory has a large number of practical applications. For example, number theory is an important member in cryptography. I'll introduce that with an example.