SESSION GRAPH BASED AND TREE METHODS + RELATED ...

48 Int'l Conf. Foundations of Computer Science | FCS'12 |
6.1 Accuracy of ALGO EXGCD
In this section we prove the accuracy of our formalization
that the functor ALGO EXGCD returns a, b and
g for given integers x and y such that ax + by = g(g
is the greatest common divisor of x and y).
We can prove the following theorem in a similarly
for proving Theorem 6.1:
Theorem 6.1 (Accuracy of ALGO EXGCD)
for x,y be Element of INT
holds
ALGO_EXGCD(x,y)‘3 = x gcd y
&
ALGO_EXGCD(x,y)‘1 * x
+ ALGO_EXGCD(x,y)‘2 * y
= x gcd y,
where ALGO EXGCD(x,y)‘n denotes the nth member
of ALGO EXGCD(x,y). Thus we proved the accuracy
of our formalization of extended Euclidean algorithm.
6.2 Multiplicative Inverse
Then, we define the functor that computes the multiplicative
inverse over a residue class ring using the
ALGO EXGCD as follows:
Definition 6.2 (Inverse)
let x,p be Element of INT;
func ALGO_INVERSE(x,p) -> Element of INT
means
for y be Element of INT
st y = (x mod p)
holds
(ALGO_EXGCD(p,y)‘3 = 1 implies
((ALGO_EXGCD(p,y)‘2 < 0) implies
(ex z be Element of INT
st z = ALGO_EXGCD(p,y)‘2
& it = p + z ))
& ( (0