Linear Congruential Generator Method | Random Numbers
Summary
TLDRThis video script delves into the Linear Congruential Generator (LCG), a popular algorithm for generating pseudo-random numbers. It explains the concept of congruence as the foundation for LCG, where a sequence of numbers is generated using a recurrence relation involving a large integer 'm', multiplier 'a', increment 'b', and a seed 'x0'. The script provides an example with specific values for 'm', 'a', 'b', and 'x0', demonstrating how to generate a sequence of pseudo-random numbers. It also touches on the generation of random bit sequences from these numbers and briefly discusses the security implications of using such sequences in cryptographic applications, hinting at the 50-50 vulnerability of the method.
Takeaways
- 🔢 The Linear Congruential Generator (LCG) is a method for generating pseudo-random numbers, which are essential for simulations and cryptography.
- 📚 The concept of congruence is fundamental to understanding LCGs, where 'a is congruent to b mod n' means n divides the difference between a and b.
- 🌰 An example of congruence is 12 being congruent to 0 mod 4, as 4 divides 12.
- 🔑 In LCGs, a large integer m, and two integers a and b are chosen, along with a seed value x₀, to generate a sequence of numbers using the formula: xi ≡ a * xi-1 + b (mod m).
- 🔄 The sequence generated by LCGs is a recurrence relation that allows for the calculation of subsequent numbers based on the previous one.
- 📈 An example provided in the script uses m=123, a=5, b=2, and x₀=73 to demonstrate the generation of pseudo-random numbers.
- 🔄 The script explains that the output of the LCG can be reduced by taking the modulo with respect to the chosen large integer m, ensuring the sequence remains within a manageable range.
- 🤔 The script mentions the possibility of generating random bit sequences from the LCG output, which can be useful for cryptographic applications requiring a stream of zeros and ones.
- 🔒 However, the script also points out that the method of generating bits from LCG is not very secure, as the predictability of the next bit is 50%, making it half secure and half vulnerable.
- 🚀 The script promises to discuss the Blum Blum Shub pseudo-random bit generator in a subsequent video, which is another method for generating random bits for cryptographic purposes.
Q & A
What is the purpose of using pseudo random numbers in simulations and cryptography?
-Pseudo random numbers are used in simulations to mimic the randomness required for various processes and in cryptography to enhance security through unpredictability, as generating true random numbers is difficult and costly.
What was the algorithm discussed in the speaker's last video for generating random numbers?
-The speaker discussed the mid square algorithm in their last video as a method for generating random numbers.
What is the mathematical concept of congruence, and how is it related to the linear congruential generator?
-Congruence is a concept in number theory where two numbers are said to be congruent modulo n if n divides their difference. It is fundamental to the linear congruential generator, which uses a recurrence relation involving multiplication, addition, and modulo operations to generate a sequence of pseudo random numbers.
What are the components of the linear congruential generator formula?
-The components of the linear congruential generator formula are a large integer m, two integers a and b, and a seed value x₀. The formula is xi ≡ a * xi-1 + b (mod m).
Why is the linear congruential generator called 'linear'?
-The linear congruential generator is called 'linear' because the recurrence relation it uses to generate numbers has a linear form in terms of the seed value and the constants a and b.
What is the significance of choosing a large integer m for the linear congruential generator?
-Choosing a large integer m for the linear congruential generator helps to increase the period of the generated sequence, making it less predictable and more suitable for applications like cryptography that require a high level of randomness.
Can you provide an example of how the linear congruential generator formula is applied?
-An example given in the script uses m=123, a=5, b=2, and x₀=73. The formula is applied iteratively to generate subsequent numbers in the sequence, such as x1 ≡ 5 * 73 + 2 (mod 123), resulting in x1=121.
How can the generated pseudo random numbers be used to create a bit sequence?
-The generated pseudo random numbers can be reduced modulo 2 to create a bit sequence. For example, if a number is odd, it would be congruent to 1 (mod 2), and if it is even, it would be congruent to 0 (mod 2).
What is the security level of the pseudo random bit sequence generated by the linear congruential generator?
-The security level of the pseudo random bit sequence generated by the linear congruential generator is considered to be 50% secure and 50% vulnerable, as the probability of correctly guessing the next bit is 50%.
What will be discussed in the speaker's next video?
-In the next video, the speaker will be discussing the Blum Blum Shub pseudo random bit generator, which is another method for generating random bits used in cryptographic applications.
Outlines
🔢 Introduction to Linear Congruential Generator
The video script begins with an introduction to the Linear Congruential Generator (LCG), a method used for generating pseudo-random numbers. The presenter explains the necessity of random numbers in simulations and cryptography, and acknowledges the impracticality and high cost of producing truly random numbers. The concept of congruence is introduced as a foundational element for understanding LCG, with examples provided to illustrate how numbers can be congruent modulo n. The script also mentions a previous video on the mid-square algorithm, another method for generating random numbers, and encourages viewers to review additional materials on congruences for a deeper understanding.
📚 Generating Pseudo-Random Numbers with LCG
This paragraph delves into the specifics of the Linear Congruential Generator algorithm. The presenter outlines the formula for generating numbers in the sequence, which involves a large integer m, multiplier a, increment b, and a seed value x0. The process of generating subsequent numbers in the sequence using the formula x_i ≡ a*x_(i-1) + b (mod m) is explained, with an example given using m=123, a=5, b=2, and x0=73. The example demonstrates how to calculate the next numbers in the sequence and emphasizes the recurrence relation that allows for the generation of a series of pseudo-random numbers.
🔑 Pseudo-Random Number Generation and Bit Sequences
The third paragraph continues the discussion on LCG, focusing on the generation of pseudo-random numbers and their application in producing bit sequences. The presenter explains how the generated numbers can be reduced modulo 2 to create a sequence of bits, which is crucial for certain cryptographic algorithms like stream ciphers. The example from the previous paragraph is used to demonstrate this process, showing how each generated number corresponds to a bit (1 for odd, 0 for even). The paragraph concludes with a brief mention of the security implications of using such a method, noting that while it is 50% secure, it is also 50% vulnerable, indicating the need for more robust methods in cryptographic applications.
Mindmap
Keywords
💡Linear Congruential Generator
💡Pseudo-random Numbers
💡Congruence
💡Seed
💡Modulus
💡Cryptography
💡Recurrence Relation
💡Random Bit Sequence
💡Stream Cipher
💡Blum Blum Shub
Highlights
Introduction to Linear Congruential Generator for generating random numbers.
Explanation of why random numbers are essential for simulations and cryptography.
Differentiation between true random numbers and pseudo random numbers due to computational feasibility.
Introduction to the concept of congruence as a foundation for the algorithm.
Definition of congruence with an example involving numbers 12 and 4.
Illustration of congruence with different numbers like 11 and 3 modulo 4.
Explanation of how congruence can be used to generate a sequence of numbers.
Description of the Linear Congruential Generator formula and its parameters m, a, b, and x0.
Example calculation using the Linear Congruential Generator with specific values for m, a, b, and x0.
Process of generating subsequent numbers in the sequence using the formula.
Discussion on the use of the generated numbers in cryptographic applications.
Generation of pseudo random numbers and their limitations in security.
Conversion of generated numbers into a bit sequence for use in cryptographic algorithms.
Analysis of the security level of the generated bit sequence and its vulnerability.
Introduction to the Blum Blum Shub pseudo random bit generator for the next video.
Summary of the Linear Congruential Generator's method and its practical applications.
Transcripts
in this video i'll be explaining linear
congruential generator for generating
random numbers
in my last video i have explained why
random numbers are required we do
require them for the various simulation
processes
and also in various cryptography
algorithms
so in my last video i have taken up
random numbers which are specifically
pseudo random numbers because it is
difficult to generate the
true random numbers and the experiment
cost us a huge
so rather than doing this we create or
we generate the random number through
certain computer algorithms
and the last algorithm that i have
talked about generating random number is
mid square algorithm now in this linear
congruential generating function
i will be requiring a concept of
congruence
which i have explained in my number
three videos i've added the link in the
description
so for an easy recall of what is
congruence
we can say that a is congruent to b
modulo n whenever n
divides a minus b so this line i will
read it as
n divides a minus b so for example
i can consider here let's take 12
and in the model i i will consider say
for example
4 so now i want to keep something here
so i can say 12 is congruent to 0 mod 4
because whenever we
subtract these two so rather we take a
minus b
or b minus a it doesn't matter because
if n divides some integer it also
divides negation of the integer
so if 4 divide 12 this is
a fine congress i can also say if i have
11 here
11 is congruent to either minus 1
because you can take minus 1 on this
side
and so this will become mod 4
so here you can say 4 divide 11
minus of minus 1 that is 12. in that
case also this congress is true
or i can also say that 11 this is
congruent to
3 mod 4 so because 11
minus 3 this is 8 and 4 divides 8.
now so this doesn't give us a unique
answer but yeah in case we want a
positive integer here so this would be
considered as a remainder whenever we
divide
this 4 by 11. so in that case you may
consider this as a unique
but we can also see a relation between
these two integer
and here we can see that minus 1 is also
congruent to 3
either you take this minus 1 on this
side it will become plus 1 or we take
3 on this side it will become minus 4
and so it is congruent to
4. so this is the normal concept of the
congruences
and there are various videos associated
with the congruences that you can look
for the further properties
now i'm going to use this type of
concept to explain what is linear
congruential generator so let m be the
large
integer i am just considering a large
integers
now for the random numbers we require
in case you are taking a cryptographic
application you do require the large
number so here i'm just taking m to be
large integers
and choose two integers a and b
so we selected two integers a and
b and i select a seed which i call it as
x naught and we defined a sequence
which sequence is related using the
congruence symbol so here we can take x
i is congruent to a times x i minus 1
plus b mod n this is something similar
as we can generate some sort of
recurrence relations or some sort of
rule which allows us to if i know one
number i can generate the other number
so as you can see
that we know the seed x naught and i can
generate the next number so
if you just fit this first seed into
this i will get the next number
so x1 this is congruent to a times x
naught plus b
times mod m and as you can see that a
and
b are already selected and we already
know once we know what is x naught we
can find what is
x 1 and similarly i can generate x 2
which is congruent to a
times x 1 plus b mod m and we continue
doing like this so it depends upon how
many integers or how many random numbers
do we require to
generate now let us take an example
corresponding to this consider this
example and in this example i have
selected let m is equal to 123
and a is equal to 5 and let b is equal
to 2 and we can choose
any random seed as well which is x
naught equal to 73
and here i will use the previous formula
we can see that x i
this is congruent to a times x i minus 1
plus b times mod m in fact this is more
like a linear congruence so that is why
the name is linear congruential method
so if somebody is very much interested
you can see my video on
what is a linear congress i've added the
link in the description
the linear congress is something like a
times x is congruent to b
mod n and there we can also talk when
this congruence is solvable or not
so this is something similar uh the only
difference is you can take
this b on the other side so it becomes a
x uh minus b
which is congruent to 0 mod n and then
we associate that for what value of x
what is the result that we get here so
in this case uh because
i'm not associating two different
numbers but here this is x i
and x i plus 1. so if you just consider
this a and
b as fixed quantity and here we consider
this as a variable
so the output that we expect here is
something which is
maybe we call it as a random number so
now once
looking this as a concept of the linear
congress extension we know what is a c
let's try to find the random numbers x 1
this is
congruent to a which is 5 and the seed
that is 73 so the first will go seed
plus 2 time which is the value for b
now solve this up which is congruent to
5 into 73 that is 365
plus 2 so this will be congruent to 367
and here we are taking modulo with
respect to 123
but now you can see this 367 is larger
than 123 so you divide
367 by 123 whatever be the remainder
so here you will get say if you are
dividing a number by
b you get some quotient plus remainder
so now we are dividing it by 123. so
whatever is the quotient
so this number a is 367 and i'm going to
get certain quotient plus the remainder
so this means
i'm going to replace this by the
remainder so that
so the remainder here is 121 so that
when this
remainder is brought here and subtracted
from here 123 will divide the difference
this is what the congress means so we
have 123.
now x1 is considered as 121.
similarly let's generate x2 number which
is 5
into now instead of this seed which is
73 i'm going to use the next number
which is generated that is 121
so we have here 121 plus 2
and again repeat the same process i am
going to first multiply 5 into 121
plus 2. so we get 607 mod
123 which is further congruent to 115
mod 123 so you can note that if i divide
607 by 123 the remainder that i get
correspondingly
is 115 so 107 minus 115 divided by 123
is i get the output as
4 so this means now i get this as a
quotient this becomes my quotient
correspondingly
so this is least remainder and similarly
we can generate
x3 which is 5 into 105 the next
seed plus 2 which is in the same format
i am just writing now the reduced number
that is 85 model of 123
and then we can generate x4 which is
congruent to 5
into 85 plus 2 which is further
congruent to 58
in the reduced list mod 123. now these
number which i have generated here
so the initial x naught that was 73
then we generated 121 then we generated
115
85 58 these are we call it as random
number and
in a specific notation you may call this
as a pseudo random number
because they are not truly random number
we have generated these through a
formula which is this
so now you may use a different linear
congruential formulas
correspondingly once you understand the
concept of generating numbers
we can use a different recurrence
relations also to generate the
random numbers now in various situation
we require random bit sequence instead
of the random numbers
so suppose that we have generated the
random number in the same order
i have taken x naught as 73 we got
x 1 which is 121 and then we got
x2 as a 115 we got
x3 as 85 and we got x4 as
58. so corresponding to this what we can
do is we can further associate
to generate the bits of the sequence
that is 0 and once
we can take the condition with respect
to modulo 2.
so if i take x naught which is 73
and 73 is congruent to 1
mod 2. so you can see that because 73 is
an odd number
so i have to so the remainder will be 1
so to get the sequence in terms of the
bits so if we further take this with
respect to modulo 2.
now in the same way if i want to
generate with respect to x1
this is 121 and 121 is also congruent to
1 mod 2.
now continue doing like this so x2 which
is 115
this is also congruent to 1 mod
2 and then x3 which is 85 and 85 is
further congruent to 1
mod 2 and then we got x 4
which is even that is 58 58 is congruent
to 0
mod 2. so had we generated certain more
numbers in this
row we could uh correspondingly generate
some more numbers in this one so what
are the random bits that we have
generated
so we got random bits as 1 1
1 1 0 and continue like this sequence
so now again uh if we want to generate
the random numbers in the bit sequence
because there are very
various crypto graphic algorithms such
as stream cipher where we do require
zeros and one
so you can generate the random bits in
this order
as well but again this is uh this is not
very secure
because we consider that uh if we look
at from the attack point of view
uh there is a possibility that the
probability of depicting the future bits
or the next bit
is half so that means this method would
be considered as 50
secure and 50 vulnerable
so but that's it for this particular
method and in the next
video i'll be talking about blum blum
shop pseudo random bit generator
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