The original version of my undergraduate research presentation that I was graded on (I got an A, but this version is certainly inferior to the later version of the presentation, by which time I also had better insight into my results).

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).