The document discusses matrix operations in Galois Fields (GF) and their significance in abstract algebra, particularly the properties of finite fields and the concept of program derivation. It covers foundational concepts such as the weakest precondition predicate transformer and various algorithms for calculating multiplicative inverses, matrix products, determinants, cofactor matrices, and matrix inverses, all within the context of GF. Additionally, it highlights the applications of matrices over GF in fields such as information theory, cryptography, and digital signal processing.

1. Program Derivation of
Matrix Operations in GF
Charles Southerland
Dr. Anita Walker
East Central University

2. The Galois Field
● A finite field is a finite set and two operators
(analogous to addition and multiplication)
over which certain properties hold.
● An important finding in Abstract Algebra is
that all finite fields of the same order are
isomorphic.
● A finite field is also called a Galois Field in
honor of Evariste Galois, a significant
French mathematician in the area of
Abstract Algebra who died at age 20.

3. History of Program Derivation
● Hoare's 1969 paper An Axiomatic Basis for
Computer Programming essentially
created the field of Formal Methods in CS.
● Dijkstra's paper Guarded Commands,
Nondeterminacy and Formal Derivation of
Programs introduced the idea of program
derivation.
● Gries' book The Science of Programming
brings Dijkstra's paper to a level undergrad
CS and Math majors can understand.

4. Guarded Command Language
This is part of the language that Dijkstra defined:
● S1
;S2
– Perform S1
, and then perform S2
.
● x:=e – Assign the value of e to the variable x.
● if[b1
S→ 1
][b2
S→ 2
]…fi – Execute exactly one of the
guarded commands (i.e. S1
, S2
, … ) whose
corresponding guard (i.e. b1
, b2
, … ) is true, if any.
● do[b1
S→ 1
][b2
S→ 2
]…od – Execute the command
if[b1
S→ 1
][b2
S→ 2
]…fi until none of the guards are
true.

5. The Weakest Precondition
Predicate Transformer wp
● Consider the mapping
wp: P⨯L → L
where P is the set of all finite-length programs and L
is the set of all logical statements about the state of
a computer.
● For S∊P and R∊L, wp(S,R) yields the “weakest” Q∊L
such that execution of S from within any state
satisfying Q yields a state satisfying R.
● With regard to this definition, we say a statement A is
“weaker” than a statement B if and only if the set of
states satisfying B is a proper subset of the set of
states satisfying A.

6. Some Notable Properties of wp
● wp([S1
;S2
],R) = wp(S1
,wp(S2
,R))
● wp([x:=e],R) = R, substituting e for x
● wp([if[b1
S→ 1
][b2
S→ 2
]…fi],R)
= (b1
∨b2
…∨ ) (∧ b1
wp(S→ 1
,R))
(∧ b2
wp(S→ 2
,R)) …∧
● wp([do[b1
S→ 1
][b2
S→ 2
]…od],R)
= (R ~b∧ 1
~b∧ 2
…∧ ) wp([if[b∨ 1
S→ 1
][b2
S→ 2
]…
fi],R)
wp([if…],wp([if…],R))∨
wp([if…],wp([if…],wp([if…],R)))∨
…∨ (for finitely many recursions)

7. The Program Derivation Process
For precondition Q∊L and postcondition R∊L, find
S∊P such that Q=wp(S,R).
● Gather as much information as possible about the
precondition and postcondition.
● Reduce the problem to previously solved ones
whenever possible.
● Look for a loop invariant that gives clues on how to
implement the program.
● If you are stuck, consider alternative
representations of the data.

8. Conditions and Background for
the Multiplicative Inverse in GF
● The precondition is that a and b be coprime
natural numbers.
● The postcondition is that x be the
multiplicative inverse of a modulo b.
● Since the greatest common divisor of a and
b is 1, Bezout's Identity yields ax+by=1,
where x is the multiplicative inverse of a.
● Recall that
gcd(a,b)=gcd(a­b,b)=gcd(a,b­a).

9. Analyzing Properties of the
Multiplicative Inverse in GF
● Combining Bezout's Identity and the given property of
gcd, we get
ax+by = gcd(a,b)
= gcd(a,b­a)
= au+(b­a)v
= au+bv­av
= a(u­v)+bv
● Since ax differs from a(u­v) by a constant multiple of
b, we get x (u­v) mod b≡ .
● Solving for u, we see u (x+v) mod b≡ , which leads
us to wonder if u and v may be linear combinations
of x and y.

10. Towards a Loop Invariant for the
Multiplicative Inverse in GF
● Rewriting Bezout's Identity using this, we get
ax+by=a(1x+0y)+b(0x+1y)
=a((1x+0y)+y­y)+b(0x+1y)
=a(x+y­y)+by
=a(x+y)­ay+by
=a(x+y)+(b­a)y
=au+(b­a)y
(so we deduce that v=y)
● Note that assigning c:=b­a and z:=x+y
would yield ax+by=az+cy.

11. Finding the Loop Invariant for the
Multiplicative Inverse in GF
● Remembering that u and v are linear combinations
of x and y, we see that by reducing the values of
a and b as in the Euclidean Algorithm gives
a1
u1
+b1
v1
=a1
(ca1x
x+ca1y
y)
+b1
(cb1x
x+cb1y
y)
=a2
(ca1x
x+ca1y
y)
+b1
((cb1x
­ca1x
)x
+(cb1y
­ca1y
)y)
= …
● After the completion of the Euclidean algorithm, we
will have gcd(a,b)(cxf
x+cyf
y)=1.

12. Algorithm for the
Multiplicative Inverse in GF
multinv(a,b) {
x:=1; y:=0
do
a>b a:=a­b; x:=x+y→
b>a b:=b­a; y:=y+x→
od
return x
}

13. C Implementation of the
Multiplicative Inverse in GF

14. Conditions and Background
of the Matrix Product in GF
● The precondition is that the number of
columns of A and the number of rows of B
are equal.
● The postcondition is that C is the matrix
product of A and B.
● The definition of the matrix product allows
the elements of C be built one at a time,
which seems to be a particularly straight-
forward approach to the problem.

15. Loop Invariant of the
Matrix Product in GF
● A good loop invariant would be that all elements of
C which either have a row index less than i or
else have a row index equal to i and have a
column index less than or equal to j have the
correct value.
● The loop clearly heads toward termination given
that C is filled from left to right, from top to
bottom (which will occur if the value of j is
increased modulo the number of columns after
every calculation, increasing i by 1 every time j
returns to 0).

16. C Implementation of the
Matrix Product in GF

17. Conditions and Background of
the Determinant of a Matrix in GF
● The precondition is that the number of rows
and the number of columns of A are equal.
● The postcondition is that d is the
determinant of A.
● The naive approach to the problem is not
very efficient, but it is much easier to
explain and produces cleaner code.

18. The Loop Invariant of the
Determinant of a Matrix in GF
● The loop invariant of the naive determinant
algorithm is that d is equal to the sum for
all k<j of the product of A1k
and the
determinant of the matrix formed by all the
elements of A except those in the first row
and kth column.
● The loop progresses toward termination so
long as the difference between the number
of columns and j decreases.

19. Conditions and Background of
the Cofactor Matrix in GF
● The precondition is that the number of rows
and the number of columns of A are equal.
● The postcondition is that C is the cofactor
matrix of A.
● Like the matrix product, the cofactor matrix
can readily be generated one element at a
time.

20. The Loop Invariant of the
Cofactor Matrix in GF
● The loop invariant of the cofactor matrix
algorithm is, like the matrix multiplication
algorithm, that for all entries in C whose
row is less than I or whose row is equal to I
and whose column is less than j.

21. Conditions and Background
of the Matrix Inverse in GF
● The precondition is that the number of rows
and the number of columns of A are equal,
and that the determinant of A be coprime
to the order of GF.
● The postcondition is that B is the matrix
inverse of A.
● Like the matrix product and the cofactor
matrix, the matrix inverse can readily be
generated one element at a time

22. Loop Invariant of the
Matrix Inverse in GF
● The loop invariant of the matrix inverse
algorithm is, like the matrix multiplication
algorithm, that for all entries in C whose
row is less than I or whose row is equal to I
and whose column is less than j.

23. Applications
● Matrices over GF have many applications
within Information Theory, including
Compression, Digital Signal Processing,
and Cryptography.
● The classic Hill cipher is a well-known
example of a use of matrix operations over
GF.
● Most modern block ciphers also use
matrices over GF, specifically the S-boxes
of ciphers like Rijndael (a.k.a. AES).