Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Prove that the RSA decryption algorithm recovers the original message; that is, medmmodpq. Hint: You may assume that, because p and q are relatively prime, it suffices to prove the congruence modp and modq.

Short Answer

Expert verified
The RSA decryption algorithm recovers the original message as medmmodpq. This is proven using the Chinese Remainder Theorem and Euler's theorem for modp and modq.

Step by step solution

01

- State the problem clearly

We need to prove that the RSA decryption algorithm recovers the original message, which means proving that medmmodpq. Given that p and q are relatively prime, it suffices to prove the congruence modp and modq.
02

- Understand RSA key structure

In RSA, e and d are the public and private exponents, respectively, and they satisfy the relationship ed1mod(p1)(q1). This means there exists an integer k such that ed=1+k(p1)(q1).
03

- Prove congruence modulo p

Consider medmodp. By Euler's theorem, since m\ot0modp and ϕ(p)=p1, it follows that mp11modp. Thus, med=m1+k(p1)=m(mp1)km1kmmodp.
04

- Prove congruence modulo q

Similarly, consider medmodq. Again, by Euler's theorem, since m\ot0modq and ϕ(q)=q1, it follows that mq11modq. Therefore, med=m1+k(q1)=m(mq1)km1kmmodq.
05

- Combine results using the Chinese Remainder Theorem (CRT)

Given that medmmodp and medmmodq, and since p and q are relatively prime, by the Chinese Remainder Theorem, these congruences imply medmmodpq.
06

Conclusion

After following these steps, it is evident that the RSA decryption algorithm correctly recovers the original message, since medmmodpq.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

RSA algorithm
The RSA algorithm is a cryptographic system used for secure data transmission. It works on the principles of public-key cryptography, where two keys are used: a public key for encryption and a private key for decryption. The process involves three key steps: key generation, encryption, and decryption. The key generation involves selecting two large prime numbers, typically denoted as p and q, then computing their product n=pq. The Euler's totient function ϕ(n) is calculated as (p1)(q1). A public exponent e is chosen such that 1<e<ϕ(n) and e is coprime to ϕ(n). The private exponent d is then determined such that ed1modϕ(n). Encryption converts a message m into a ciphertext c using the formula c=memodn. Decryption reverts the ciphertext c back to the original message m using m=cdmodn.
Number theory
Number theory is a branch of pure mathematics dealing primarily with integers and integer-valued functions. It involves properties and relationships of numbers, especially the more complex aspects like divisibility, primes, and the solutions of equations in integers. Within the context of the RSA algorithm, number theory provides foundational principles such as prime factorization and the properties of modular arithmetic. Understanding number theory helps in comprehending why certain encryption methods work and establishes the security of algorithms like RSA, which rely on the difficulty of factoring the product of two large primes.
Chinese Remainder Theorem
The Chinese Remainder Theorem (CRT) is a result from number theory that provides a way to solve systems of simultaneous congruences with different moduli. It essentially states that if one knows the remainders of the division of an integer by several pairwise coprime integers, then one can uniquely determine the remainder of the division of that integer by the product of these integers. In the context of RSA decryption, CRT is used to combine congruences. Since we prove the congruence separately mod p and mod q, CRT allows us to conclude that the results hold mod pq. This theorem is crucial for simplifying and speeding up the RSA decryption process.
Euler's theorem
Euler's theorem is a key result in number theory that states: if a and n are coprime, then aϕ(n)1modn. Euler's totient function ϕ(n), counts the number of integers up to n that are relatively prime to n. In RSA, Euler's theorem plays a crucial role as it guarantees the existence of a multiplicative inverse needed for key generation. When we raise the message m to the power of ed, Euler's theorem helps us show that medmmodp and medmmodq. Essentially, Euler's theorem ensures the consistency and correctness of the RSA decryption process.
Modular arithmetic
Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after reaching a certain value, known as the modulus. In RSA encryption and decryption, modular arithmetic is used to ensure that integers do not exceed the bounds set by the modulus, typically n=pq. It involves operations like modular addition, subtraction, multiplication, and exponentiation. To decrypt the message in RSA, we rely on properties of modular exponentiation: computing powers of numbers modulo n. Hence, modular arithmetic is foundational to the functioning of the RSA algorithm, helping in keeping calculations manageable even with large numbers.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose that at round i in DES, Li1 is all 0 s, Ri1 is (in hex) deadbeef, and Ki is a5bd96 860841 . Give Ri, assuming that we use a simplified S box that reduces each 6-bit chunk to 4 bits by dropping the first and last bits.

Suppose you are doing RSA encryption with p=101,q=113, and e=3. (a) Find the decryption exponent d. (Hint: Although there are methodical ways to do this, trial and error is efficient for e=3. ) (b) Encrypt the message m=9876. Note that evaluating m3 with 32 -bit arithmetic results in overflow.

Suppose you want your filter-based firewall to block all incoming Telnet connections, but to allow outbound Telnet connections. One approach would be to block all inbound packets to the designated Telnet port (23). (a) We might want to block inbound packets to other ports as well, but what inbound TCP connections must be permitted in order not to interfere with outbound Telnet? (b) Now suppose your firewall is allowed to use the TCP header Flags bits in addition to the port numbers. Explain how you can achieve the desired Telnet effect here while at the same time allowing no inbound TCP connections.

Suppose a firewall is configured to allow outbound TCP connections but inbound connections only to specified ports. The FTP protocol now presents a problem: When an inside client contacts an outside server, the outbound TCP control connection can be opened normally but the TCP data connection traditionally is inbound. (a) Look up the FTP protocol in, for example, Request for Comments 959 . Find out how the PORT command works. Discuss how the client might be written so as to limit the number of ports to which the firewall must grant inbound access. Can the number of such ports be limited to one? (b) Find out how the FTP PASV command can be used to solve this firewall problem.

Estimate the probabilities of finding two messages with the same MD5 checksum, given total numbers of messages of 263,264, and 265. Hint: This is the birthday problem again, as in Exercise 49 of Chapter 2, and again the probability that the k+1 th message has a different checksum from each of the preceding k is 1k/2128. However, the approximation in the hint there for simplifying the product fails rather badly now. So, instead, take the log of each side and use the approximation log(1k/2128)k/2128.

See all solutions

Recommended explanations on Computer Science Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free