y solution to this equation is r = 1, since 5 X 1 mod 3 = 5 mod 3 = 2. We, thus, see that |11/5/3 = 1, which is not at all intuitive. One can also define the number (11/5) mod 3 as the product (11 X 1/5) mod 3 or, equivalently, (11 mod 3) X (1/5 mod 3) = 2 X (1/5 mod 3) mod 3. According to the same procedure, we find that |1/5/3 = 2, since 5 x 2 = 10 mod3 = 1. We, thus, obtain 2 X (1/5 mod 3) mod 3 = 2 x 2 mod 3 = 4 mod 3 = 1 = (11/5) mod 3, which is consistent with the previous definition of division. Since inverses can be defined in modular arithmetic, there is no need to use the division operation. However, a restriction applies to the relation between the number to be inverted and the modulus: they should be co-prime, meaning that they should have one as their greatest common divider (see further). The conversion between x and its residue r (modulo m) is called modular reduction. Such a reduction operation corresponds to a one-way function. This term refers to the fact that the correspondence that is established is not reversible, since there is an infinity of numbers with the same residue. Powers and exponentials of the type y = xP and y = ax (a, p = integers) are also one-way functions, since the result y is associated with an infinity of possible values X. Here is an important property concerning the commutation of exponents, which is the same as in ordinary arithmetic, and which we will have the opportunity to exploit: (aP mod m) mod m = (a9 mod m)P mod m = a mod m. (T12) Appendix T Having introduced modular arithmetic basics, I can describe two fundamental theorems that have been derived by Fermat and Euler. Two numbers are said to be relatively prime, or co-prime to each other, if they do not divide each other, or if their greatest common divider (GCD) is unity. For instance, (3,4), (2,9), and (5,12) are co-prime. Clearly, two prime numbers are co-prime. Primes and co-primes play a central role in modular arithmetic, as the Fermat and Euler theorems and applications illustrate. Fermat's theorem states t