Q: Step 1: Generating the first number ($X_1$)

We plug our seed ($X_0 = 7$) into the formula: $X_1 = (5 \times 7 + 3) \pmod{16}$ $X_1 = (35 + 3) \pmod{16}$ $X_1 = 38 \pmod{16}$ To find $38 \pmod{16}$, we divide 38 by 16. 16 goes into 38 two times ($16 \times 2 = 32$), leaving a remainder of 6. Result: Our first random number is 6.

Question 1

Step 1: Generating the first number ($X_1$)

Accepted Answer

We plug our seed ($X_0 = 7$) into the formula:
$X_1 = (5 	imes 7 + 3) \pmod{16}$
$X_1 = (35 + 3) \pmod{16}$
$X_1 = 38 \pmod{16}$
To find $38 \pmod{16}$, we divide 38 by 16. 16 goes into 38 two times ($16 	imes 2 = 32$), leaving a remainder of 6.
Result: Our first random number is 6.

Question 2

Step 2: Generating the second number ($X_2$)

Accepted Answer

We take our previous result ($X_1 = 6$) and plug it back into the formula:
$X_2 = (5 	imes 6 + 3) \pmod{16}$
$X_2 = (30 + 3) \pmod{16}$
$X_2 = 33 \pmod{16}$
Divide 33 by 16. It goes in twice (32), leaving a remainder of 1.
Result: Our second random number is 1.

Question 3

Step 3: Generating the third number ($X_3$)

Accepted Answer

We take our previous result ($X_2 = 1$) and plug it in:
$X_3 = (5 	imes 1 + 3) \pmod{16}$
$X_3 = (5 + 3) \pmod{16}$
$X_3 = 8 \pmod{16}$
Divide 8 by 16. It goes in zero times, leaving a remainder of 8.
Result: Our third random number is 8.

Question 4

Can a random number generator be hacked?

Accepted Answer

Yes, but the method depends entirely on the type of generator. A standard PRNG (like those used in basic video games) can be trivially "hacked" if an attacker observes a few output numbers, reverse-engineers the mathematical formula, and calculates the internal state to predict all future numbers. A Cryptographically Secure PRNG cannot be mathematically predicted in this way. However, CSPRNGs can be hacked if the attacker compromises the physical hardware (e.g., reading the computer's memory to steal the seed) or if the software developer implements the generator poorly, such as failing to gather sufficient entropy during the system boot process.

Question 5

What is a seed, and why do we need it?

Accepted Answer

A seed is the initial numerical value fed into a pseudo-random number algorithm. Because computer algorithms are strictly deterministic (they follow rules exactly), they cannot create randomness out of thin air. They must have a starting point. The algorithm takes the seed and scrambles it mathematically to produce the first random number. We need seeds because they dictate the entire subsequent chain of numbers. If you know the seed, you know every number the generator will ever produce.

Question 6

Why does my random number generator output the same sequence every time?

Accepted Answer

This occurs because you are using a pseudo-random number generator and you are providing it with the exact same initial seed value every time your program runs. Many programming languages require you to explicitly initialize the generator with a changing value. To fix this, developers typically seed their PRNG using the current system time (e.g., the number of milliseconds elapsed since January 1, 1970). Because the exact millisecond you run the program is always different, the seed is different, resulting in a unique sequence of numbers on every execution.

Question 7

Is the Mersenne Twister cryptographically secure?

Accepted Answer

No, the Mersenne Twister is absolutely not cryptographically secure. Despite its massive period and excellent statistical distribution, its internal mathematical transformations are linear and easily reversible. If an attacker observes exactly 624 consecutive outputs from a standard MT19937 generator, they have enough information to completely reconstruct the algorithm's internal state array. From that moment on, they can perfectly predict every subsequent number the generator will produce. It should never be used for passwords, tokens, or encryption.

Question 8

How do hardware random number generators work?

Accepted Answer

Hardware True Random Number Generators (TRNGs) work by measuring microscopic physical phenomena that are inherently unpredictable according to the laws of physics. A common method involves measuring thermal noise: the random microscopic vibrations of electrons inside a semiconductor resistor caused by heat. An analog-to-digital converter measures the tiny voltage fluctuations caused by these vibrating electrons and translates those fluctuations into digital ones and zeros. Because the movement of the electrons is governed by quantum mechanics, the resulting numbers are fundamentally impossible to predict.

Question 9

What is modulo bias?

Accepted Answer

Modulo bias is a statistical flaw introduced when a programmer uses the modulo operator (%) to force a large random integer into a smaller range, and the maximum value of the random integer is not evenly divisible by the desired range. This results in the lower numbers in the target range appearing slightly more frequently than the higher numbers. While the bias might be mathematically tiny (e.g., a fraction of a percent), in high-volume applications like online poker or cryptography, this slight statistical edge can be exploited by attackers to break the system or gain an unfair advantage. It is mitigated by using rejection sampling.

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