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In this YouTube video on Class4crypt c4c10.6a Cifrado y descifrado con RSA (parte 1), the speaker discusses the RSA algorithm, its encryption and decryption processes, and its application for digital signatures. Prime numbers are chosen, multiplied to obtain the modulo, and a public key is established that satisfies the multiplicative inverses condition. Private keys can be computed using the extended Euclidean algorithm or the Carmichael function. The algorithm is slow compared to symmetric cryptography, making it primarily used in educational or laboratory settings or for encrypting small secret numbers. The next video will explore how to apply encryption and digital signatures simultaneously using RSA.

**00:00:00**In this section, the teacher introduces the agenda for the day and starts with a recap of the RSA algorithm, where each user chooses a pair of distinct primes and multiplies them to obtain the modulo used for encryption. They then look for a public key that satisfies the multiplicative inverses condition and compute the private key using the extended Euclidean algorithm or the Carmichael function. The teacher notes that it is recommended to keep the primes secret in case of large computational costs, and for encryption, the plaintext is reduced to the public modulo while the ciphertext is raised to the inversely used exponent at the receiver end.**00:05:00**this section, the video explains the process of encrypting and decrypting with the RSA algorithm. The ciphertext is obtained by raising the number to be encrypted to a certain exponent which will either be the public key in case of confidential transmission or private key in case of digital signing, and reducing it modulo n, which is a public module. The video also explains that in the case of simultaneous encryption and signing, one needs to be careful with the size of the numbers to be used to ensure that what is encrypted or signed is still within the module size. It also notes that the RSA algorithm is very slow compared to symmetric cryptography and is mainly used to encrypt small secret numbers or in educational or laboratory settings.**00:10:00**In this section, the video discusses the use of the inverse key in decrypting messages using RSA encryption. The video demonstrates that if a specific condition is met, which is if the number being decrypted is a remainder of the modulus of the public key, then the message can be decrypted by the receiver using the inverse of the key used by the sender. The condition is proven using Euler's theorem for prime numbers and the Chinese Remainder Theorem. However, the method only works for a limited set of numbers, and the video suggests referring to their previous lesson on the Chinese Remainder Theorem for a more comprehensive discussion.**00:15:00**In this section, the speaker discusses the theorem of Euler and the Chinese remainder theorem to demonstrate the validity of the system for all values. They use a software tool to compute exponential values and check both modular exponentiation operations of both theorems. They observe that for some numbers, the value is equal to 1, which passes the Euler theorem. However, they still face issues with the numbers having common factors with pq. Thus, they move ahead and check the Chinese remainder theorem with a different exponent to verify it.**00:20:00**In this section of the video, the instructor goes through an example of RSA encryption and decryption using the Chinese Remainder Theorem. He chooses two prime numbers and a number to be encrypted, and calculates the public key, private key, and encrypted value. He emphasizes the importance of using modular arithmetic throughout the computation.**00:25:00**In this section of the video, the speaker explains how to decrypt a message using RSA encryption by using the private key. The speaker demonstrates how, by using the private key, anyone can decrypt a message that has been encrypted with their public key. The speaker also discusses the use of the Chinese remainder theorem to optimize the time for the decryption process. The section concludes with a discussion of the importance of using large numbers for both the private and public keys in order to ensure greater security.**00:30:00**In this section, the instructor explains the process of encryption and decryption with integrity and authenticity using RSA, which is known as digital signature. The sender first applies a hash function to the document and then encrypts it with their private key, reducing it to the receiver's public key module. The receiver uses their private key to decrypt the cryptogram and then applies the same hash function to the document to verify its integrity and to check the identity of the sender. The instructor also discusses the limitations and practicality of using RSA for conventional encryption, as well as the importance of third-party trust authorities for verifying identities in digital signatures.**00:35:00**In this section, the creator demonstrates how to encrypt and decrypt a document with RSA encryption. They use two primes of 12 bits each to create a public key that is as small as possible. The creator demonstrates how the document, in this case the number 2001, is encrypted with the recipient's public key and decrypted with the creator's private key. They also show how the same process can be used for digital signatures, with the private key used for signing and the public key used for verification. The creator also provides a demonstration of how to use the software GenereSEA for RSA encryption and decryption.**00:40:00**In this section, the video covers the basics of generating keys, encryption and decryption using RSA, and digital signatures. It emphasizes that in asymmetric cryptography, only numbers are encrypted, not messages, text, or files. Through demonstrations, the audience learns about necessary operations, such as Euler's totient function and reduced residue systems, are used to perform encryption and decryption. The video also explores how RSA is primarily used for key exchange and digital signatures, not for the encryption of messages. Finally, the next video is previewed, which will explore how to apply encryption and digital signatures simultaneously using RSA, and highlights the importance of cipher block size.**00:45:00**In this section, the instructor provides some recommended reading for those who want to further explore the topics covered in the video. He shares a link to a paper that details the Chinese Remainder Theorem, which will be covered in a future class, as well as a webpage with comprehensive information on cryptography. The instructor also mentions a well-known website that provides simple examples of encryption and decryption, and promises to cover this topic with more advanced methods in a future class. He ends the video by thanking viewers for their time, and directing them to his YouTube channel and Twitter for more information on his project.

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