Number theory and RSA Essay Example | Topics and Well Written Essays - 3250 words

Number theory and RSA - Essay Example At this stage, the message is sent to the recipient, for the receiver to get the clear message, the message is decrypted back to original plaintext. The whole process usually rely on a key, this key give a way to encrypt the message by the sender and a way for the listener to figure out what the message says (decrypt). This can be enabled with the use of the old cryptosystem which used the symmetric keys or the newly and more advanced public key cryptosystem. In symmetric keys one had to possess the key that was used to encrypt the message so as to decrypt back the message at the other end; but for the public key cryptosystem knowing the message does not mean one can decrypt the message (Stinson, 2006). For example, the trapdoor function; in this idea, a function ? is for encryption, so, to decrypt one has to find the inverse, ; which is difficult to find, this means that even if you have ? and you understand the output is A, it is still hard to find (A). However, if it happens that you know k, (extra piece information that acts as a lever that lets the inverse fall through by opening the trapdoor), then from ?, A and k one can easily come up with (A), so in this system k is kept secret but the whole message can be displayed or heard by anyone and it does not pose any threat because nobody can decrypt the message. Number Theory In Number theory, theorems have been derived to explain the public key cryptosystem and the key exchange problem (Martin J. Erickson, 2008). Some studies introduce the sage commands to aid in performing basic number theorems such as Euler’s phi function and greatest common divisor. This happens when coding the algorithms and functions, but mathematically it is not a requirement as we will notice in the examples. In definition, integers greater than 1, and their positive divisors are only 1 and itself, then that is a prime number, otherwise if not then it is a composite number (Martin J. Erickson, 2008). The fundamental theorem of arithmetic is the base equation when dealing with number theory is; n The greatest common divisor of k and m, (gcd (k, m)), is the largest number that can divide both k and m. this two positive integers greater than 0 are relatively prime if gcd (k, m) =1. This can be shown by the extended Euclidian algorithm below (T. H. Cormen, 2001), Input: k, m Output: (d, x, y) subject to d=gcd(k,m) and kx + my = d d=k; t=m; x=1; y=0; r=0; s=1; While (t>0) { q = _d/t_ u=x-qr; v=y-qs; w=d-qt x=r; y=s; d=t r=u; s=v; t=w } return (d, x, y) In the above algorithm, these Invariants were used gcd(k,m)=gcd(d,t) kx + my = d kr + ms = t using the Chinese Reminder Theorem(CRT) (Loxton, 1990), we Let m1, m2, ,,, mk be integers subject to gcd(mi, mj) = 1, i j, and †¦ This means that there exist a unique solution modulo m= and this solution is given by (a1, a2,†¦ ak) = (? ) mod m, Where = m /, and = mod This idea can be proven by these four steps; 1. Set M = 2. Find such that + = 1(this is a Bezou t). 3. Note that = 1 if j 4. Therefore, the solution to the system equation becomes: Mentioned above is the Bezout’s Identity, which is a property in the greatest common divisor (Martin J. Erickson, 2008). The Bezout’s identify states that for any two natural numbers k and m, there exist integers x