The Extended Euclidean Algorithm (EEA) is an extension of the Euclidean Algorithm, which is a classical method for finding the greatest common divisor (GCD) of two integers. While the standard Euclidean Algorithm merely computes the GCD, the Extended Euclidean Algorithm also determines how this GCD can be expressed as a linear combination of the original two integers. This additional capability of EEA is particularly significant in various cryptographic applications, such as finding modular inverses, which are essential for public-key cryptography.
The Euclidean Algorithm
The Euclidean Algorithm is based on the principle that the GCD of two integers and
(where
) is the same as the GCD of
and
. This can be expressed as:
The algorithm proceeds by repeatedly applying this principle until the remainder is zero. The last non-zero remainder is the GCD of the original two numbers.
Example:
Let us find the GCD of 252 and 198 using the Euclidean Algorithm.
1. with a remainder of
(i.e.,
)
2. with a remainder of
(i.e.,
)
3. with a remainder of
(i.e.,
)
4. with a remainder of
(i.e.,
)
Since the remainder has reached zero, the GCD is the last non-zero remainder, which is .
The Extended Euclidean Algorithm
The Extended Euclidean Algorithm not only finds the GCD of two integers but also finds coefficients and
such that:
These coefficients and
are known as the Bézout coefficients. The process involves back-substitution of the remainders obtained during the execution of the Euclidean Algorithm.
Example:
Using the same numbers, ![Rendered by QuickLaTeX.com 252](https://eitca.org/wp-content/ql-cache/quicklatex.com-dc6371c9313c2b2cc31629962180c159_l3.png)
![Rendered by QuickLaTeX.com 198](https://eitca.org/wp-content/ql-cache/quicklatex.com-06a62ed2a850276b6fd2296a6ea43275_l3.png)
![Rendered by QuickLaTeX.com x](https://eitca.org/wp-content/ql-cache/quicklatex.com-7e5fbfa0bbbd9f3051cd156a0f1b5e31_l3.png)
![Rendered by QuickLaTeX.com y](https://eitca.org/wp-content/ql-cache/quicklatex.com-38461fc041e953482219abf5d4cce1cb_l3.png)
![Rendered by QuickLaTeX.com 252x + 198y = 18](https://eitca.org/wp-content/ql-cache/quicklatex.com-b9de0745d330934003f83b96038c3bf9_l3.png)
From the Euclidean Algorithm steps:
1.
2.
3.
4.
We now back-substitute to express as a combination of
and
:
From step 3:
From step 2:
Substitute this into the equation for :
From step 1:
Substitute this into the equation for :
Thus, and
are the coefficients such that:
Significance in Cryptographic Applications
One of the critical applications of the Extended Euclidean Algorithm in cryptography is finding the modular inverse. In public-key cryptosystems like RSA, modular arithmetic is extensively used. The modular inverse of an integer modulo
is an integer
such that:
This means that leaves a remainder of
when divided by
. The modular inverse exists if and only if
and
are coprime (i.e.,
).
The Extended Euclidean Algorithm can be used to find this inverse. Given and
, the EEA computes integers
and
such that:
Taking this equation modulo , we get:
Thus, is the modular inverse of
modulo
.
Example:
Consider finding the modular inverse of ![Rendered by QuickLaTeX.com 3](https://eitca.org/wp-content/ql-cache/quicklatex.com-ce2009a45822333037922ccca0872a55_l3.png)
![Rendered by QuickLaTeX.com 11](https://eitca.org/wp-content/ql-cache/quicklatex.com-ef822489b9748c10966e5e94b8463f3a_l3.png)
Using the Extended Euclidean Algorithm:
1.
2.
3.
Back-substitute to find the Bézout coefficients:
Substitute into the equation for :
Thus, is the modular inverse of
modulo
, since:
Conclusion
The Extended Euclidean Algorithm is a powerful tool in number theory and cryptography, providing not only the GCD of two integers but also the coefficients needed to express this GCD as a linear combination of the original integers. This capability is crucial in cryptographic applications, particularly for finding modular inverses, which are essential for the functioning of public-key cryptosystems like RSA. By leveraging the EEA, cryptographers can ensure the mathematical underpinnings of encryption and decryption processes are robust and secure.
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