Randu Prng

RANDU is a pseudorandom number generator from the early 1960s. It is a linear congruential generator, specifically a Lehmer random number generator.

Formula

RANDU has a very short formula. Starting from the seed value, you can calculate the next value using this formula.

LCG parameters

As RANDU is a linear congruential generator (LCG), it can be defined by its LCG parameters.

Recommendations

It is highly discouraged to use RANDU in any new projects. Its flaws have been known for a very long time and there are many alternatives that are superiour in every way.

Python implementation

In [2]:

def RANDU(n):
    n *= 65539
    n %= 0x80000000
    return n

In [3]:

RANDU(1)
RANDU(RANDU(1))
RANDU(RANDU(RANDU(1)))

Out [3]:

In [4]:

RANDU = LCG(2**31, 2**16 + 3, 0)

In [5]:

RANDU(1)
RANDU(RANDU(1))
RANDU(RANDU(RANDU(1)))

Out [5]:

Analysis

In this section, I will try to visually inspect the output of RANDU in order to see how obvious its flaws are. We will inspect 1-D, 2-D, and 3-D outputs in that order.

Due to the known flaws of RANDU, 3-D outputs should completely give away how broken it is. But let’s go through all the steps and see if we can identify any signs earlier.

1-D analysis

In [6]:

vals = []

RANDU.seed_urandom()
for _ in range(2_000_000):
    x = 0
    x += RANDU.next_float()
    x += RANDU.next_float()
    x += RANDU.next_float()
    vals.append(x - 1.5)

_ = plt.hist(vals, bins=200)

In [7]:

RES = 1024

RANDU.seed_urandom()
vals = [RANDU.next_float() for _ in range(RES * RES)]
vals = np.array(vals).reshape((RES, RES))

_ = plt.imshow(vals, cmap=cm)

In [8]:

RES = 1024

vals = [0 for _ in range(RES * RES)]

RANDU.seed_urandom()
for _ in range(10_000_000):
    index = int(RANDU.next_float() * (RES * RES))
    vals[index] += 1

vals = np.array(vals).reshape((RES, RES))

_ = plt.imshow(vals, cmap=cm)

In [9]:

RES = 512

vals = [0 for _ in range(RES * RES)]

RANDU.seed_urandom()
for _ in range(200):
    for index in range(RES * RES):
        vals[index] += RANDU.next_float()

vals = np.array(vals).reshape((RES, RES))

_ = plt.imshow(vals, cmap=cm)
_ = plt.colorbar()

2-D analysis

In [10]:

RES = 256

vals = np.zeros((RES, RES))

RANDU.seed_urandom()
for _ in range(10_000_000):
    x = int(RANDU.next_float() * RES)
    y = int(RANDU.next_float() * RES)
    vals[x][y] += 1

_ = plt.imshow(vals, cmap=cm)
_ = plt.colorbar()

In [11]:

RES = 2**11

vals = [0 for _ in range(RES * RES)]

RANDU.seed_urandom()
for _ in range(5_000_000):
    index = int(RANDU.next_float() * (RES * RES))
    vals[index] += RANDU.next_float()

vals = np.array(vals).reshape((RES, RES))

_ = plt.imshow(vals, cmap=cm)

3-D analysis

In [12]:

RES = 1024
values = np.zeros((RES, RES))

RANDU.seed_urandom()
for _ in range(10_000_000):
    x = RANDU.next_float()
    y = RANDU.next_float()
    
    x = int(x * RES)
    y = int(y * RES)
    values[y][x] += RANDU.next_float()

_ = plt.imshow(values, cmap=cm)

In [13]:

x = []
y = []
z = []

RANDU.seed_urandom()
for i in range(100_000):
    x.append(RANDU.next_float())
    y.append(RANDU.next_float())
    z.append(RANDU.next_float())

fig = plt.figure()
ax = fig.add_subplot(projection='3d')
ax.azim = 0
ax.elev = 0
_ = ax.scatter(x, y, z, s=0.05)

In [14]:

x = []
y = []
z = []

RANDU.seed_urandom()
for i in range(100_000):
    x.append(RANDU.next_float())
    y.append(RANDU.next_float())
    z.append(RANDU.next_float())

fig = plt.figure()
ax = fig.add_subplot(projection='3d')
ax.azim = 57
ax.elev = 0
_ = ax.scatter(x, y, z, s=0.05)