Problem Statement

The linear congruential generator is a very simple example of a random number generator. All linear congruential generators use this formula:

Where:

If one chooses the values of

a

{\displaystyle a}

,

c

{\displaystyle c}

and

m

{\displaystyle m}

with care, then the generator produces a uniform distribution of integers from

0

{\displaystyle 0}

to

m − 1

{\displaystyle m-1}

. LCG numbers have poor quality.

r

n

{\displaystyle r_{n}}

and

r

n + 1

{\displaystyle r_{n+1}}

are not independent, as true random numbers would be. Anyone who knows

r

n

{\displaystyle r_{n}}

can predict

r

n + 1

{\displaystyle r_{n+1}}

, therefore LCG is not cryptographically secure. The LCG is still good enough for simple tasks like Miller-Rabin primality test, or FreeCell deals. Among the benefits of the LCG, one can easily reproduce a sequence of numbers, from the same

r

0

{\displaystyle r_{0}}

. One can also reproduce such sequence with a different programming language, because the formula is so simple. The task is to replicate two historic random number generators. One is the rand() function from BSD libc, and the other is the rand() function from the Microsoft C Runtime (MSCVRT.DLL). Each replica must yield the same sequence of integers as the original generator, when starting from the same seed. In these formulas, the seed becomes

s t a t

e

0

{\displaystyle state_{0}}

. The random sequence is

r a n

d

1

{\displaystyle rand_{1}}

,

r a n

d

2

{\displaystyle rand_{2}}

and so on.

The BSD formula was so awful that FreeBSD switched to a different formula. More info is at Random number generator (included)#C.

Let's start with the solution:

Program LinearCongruentialGenerator(output);
{$mode iso} 
var
  x1, x2: int64;
 
function bsdrand: cardinal;
  const
    a = 1103515245;
    c = 12345;
    m = 2147483648;
  begin
    x1 := (a * x1 + c) mod m;
    bsdrand := x1;
  end;
 
function msrand: cardinal;
  const
    a = 214013;
    c = 2531011;
    m = 2147483648;
  begin
    x2 := (a * x2 + c) mod m;
    msrand := x2 div 65536;
  end;
 
var
  i: cardinal;
begin
  writeln('      BSD            MS');
  x1 := 0;
  x2 := 0;
  for i := 1 to 10 do
    writeln(bsdrand:12, msrand:12);
end.