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Lec 5 | MIT 6.042J Mathematics for Computer Science, Fall 2010

MIT OpenCourseWare・44 minutes read

The lecture covers encryption using number theory, with Turing proposing secure schemes involving primes and keys for encryption and decryption. Euler's totient function, RSA encryption, and the proof of Euler's theorem are discussed, highlighting the importance of mathematical concepts in understanding encryption algorithms.

Insights

Encryption in number theory involves transforming plain text into encrypted text using keys shared between sender and receiver, while decryption reverses this process with the same keys.
Turing's code version two enhances security by using a public prime and secret prime for encryption, requiring finding the multiplicative inverse of the secret key modulo the public prime for decryption.
Euler's theorem, essential for RSA encryption, establishes a congruence relationship between remainders and integers relatively prime to n, proving the profound link between prime numbers, remainders, and the Euler's totient function.

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Recent questions

What is encryption in number theory?
Encryption involves transforming plain text into encrypted text.
How does decryption work in cryptography?
Decryption reverses encryption to reveal the original message.
What is the significance of Euler's totient function?
Euler's totient function counts integers relatively prime to n.
How does RSA encryption ensure security?
RSA encryption uses public and secret keys for secure communication.
What is the role of remainders in Euler's theorem?
Remainders play a key role in proving Euler's theorem.

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Summary

00:00

Turing's Encryption: Enhancing Security with Number Theory

The lecture on number theory covers encryption as an application of number theory.
Encryption involves sharing keys between sender and receiver to transform plain text into encrypted text using an algorithm.
Decryption reverses the process using the same keys to transform encrypted text back into plain text.
Turing proposed using number theory in cryptography, starting with a scheme called Turing's code version one.
Turing's code version one involves translating a message into a prime number and encrypting by multiplying with a secret prime key.
However, this scheme is insecure as intercepting two encrypted messages allows for key retrieval by finding the greatest common divisor.
To enhance security, Turing's code version two involves exchanging a public prime and a secret prime for encryption.
Encryption in version two involves taking the remainder after dividing the product of the message and secret key by the public prime.
Decryption in version two requires finding the multiplicative inverse of the secret key modulo the public prime.
The lecture delves into modular arithmetic, defining congruency and the concept of multiplicative inverses to enhance encryption security.

17:59

Understanding Multiplicative Inverses and RSA Algorithm

The multiplicative inverse of x modulo n is a number denoted as x^-1, ranging from 0 to n-1, such that x times x^-1 is congruent to 1 modulo n.
An example is given with 2 times 3 equaling 6, congruent to 1 modulo 5, showing that 2 is the multiplicative inverse of 3 modulo 5.
Another example is shown with 5 times 5 equaling 25, congruent to 1 modulo 6, demonstrating that 5 is its own multiplicative inverse modulo 6.
The concept of the remainder of m times k after dividing out multiples of p is discussed, with the remainder congruent to m times k modulo p.
Encryption is shown to be congruent to the plain message times the key modulo p, with decryption using the multiplicative inverse of the key.
The decryption process involves finding the multiplicative inverse of the key modulo p and using it to decrypt the message.
The known plaintext attack is explained, where knowing a plaintext message and its corresponding encrypted message can lead to breaking the encryption scheme.
Euler's totient function, denoted as phi of n, counts the integers up to n-1 that are relatively prime to n.
Examples are given with n=12 and n=15, showing how to calculate Euler's totient function for different values of n.
The text concludes by mentioning the significance of Euler's totient function in understanding the RSA algorithm and its role in encryption.

36:48

Euler's Theorem: Relating Remainders and Primes

13 and 14 are relatively prime numbers, with 13 being a prime number and 14 being the product of 2 and 7, and thus Euler's totient function evaluated at 15 equals 8.
The Euler's totient function can be easily calculated if the prime factorization of a number is known, and this information can be found in a book.
Euler's theorem states that if the greatest common divisor of two numbers is 1, then a specific congruence relationship holds, leading to Fermat's little theorem and applications like the RSA algorithm.
A lemma is introduced to prove Euler's theorem, showing that if the greatest common divisor of two numbers is 1, then their remainders after division by n are relatively prime to n.
Another lemma is presented, demonstrating that the set of remainders after division by n is equal to the set of integers relatively prime to n, with the number of elements in the set being equal to the Euler's totient function evaluated at n.
The first part of the proof shows that all remainders in the set are distinct, leading to the conclusion that there are exactly r (Euler's totient function value) remainders.
The second part of the proof uses properties of greatest common divisors to show that all remainders are relatively prime to n, establishing that the set of remainders is a subset of the set of integers relatively prime to n.
The proof of Euler's theorem involves using the second lemma to show that the product of all remainders is congruent to the product of all integers relatively prime to n modulo n.
By leveraging the properties of greatest common divisors and the lemma, Euler's theorem is proven, showcasing the relationship between remainders and integers relatively prime to n.
The process of proving Euler's theorem involves intricate mathematical steps and lemmas, culminating in a profound understanding of the relationship between remainders, prime numbers, and the Euler's totient function.

56:17

"RSA Encryption: Public Key Security Explained"

The relationship between sets k1, k2, ..., kr is explored, showing congruence to remainders after dividing by n.
Each remainder k1k, k2k, ..., krk modulo n is shown to be congruent to k1k, k2k, ..., krk modulo n.
The product k1k, k2k, ..., krk modulo n is regrouped to reveal a pattern.
The product of relatively prime numbers k1, k2, ..., kr is shown to be relatively prime with respect to n.
The multiplicative inverse of the product is used to prove an equation involving Euler's totient function.
RSA encryption is introduced as a public key encryption scheme.
The process of generating public and secret keys for RSA encryption is detailed.
Encryption involves raising the message to the power of e modulo n.
Decryption involves raising the encrypted message to the power of d modulo n.
The decryption process is proven using Fermat's theorem and congruency properties.

01:15:12

RSA Encryption: Enduring Effectiveness and Recent Advancements

If m is congruent to 0 modulo p, then it equals 0, which holds true for any case. Similarly, q divides the difference of m prime d minus m, leading to the conclusion that p times q divides m prime d minus m, which is equal to n, resulting in m prime to the power d being congruent to m modulo n.
The decryption rule for RSA encryption involves finding the remainder of m prime to the power d after multiplying out as many multiples of n as possible, showcasing the enduring effectiveness of RSA encryption despite recent advancements in cryptography, such as Craig Gentry's breakthrough in evaluating Boolean circuits under encryption.

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