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4.4. Suppose that Alice and Bob communicate using the RSA PKC. This means that Alice has a public modulus NA = PAqA, a public encryption exponent eд, and a private decryption exponent dд, where PA and q₁ are primes and eд and dд satisfy eAdA 1 (mod (PA - 1)(A-1)). Similarly, Bob has a public modulus NB = PBqв, a public encryption expo- nent eg, and a private decryption exponent dB. In this situation, Alice can simultaneously encrypt and sign a message in the following way. Alice chooses her plaintext m and computes the usual RSA ciphertext c = mB (mod NB). She next applies a hash function to to her plaintext and uses her private decryption key to compute s = Hash (m)dA (mod N₁). She sends the pair (c, s) to Bob. Bob first decrypts the ciphertext using his private decryption exponent dB, mcdB (mod NB). = He then uses Alice's public encryption exponent eд to verify that Hash (m) SA (mod NA). Explain why verification works, and why it would be difficult for anyone other than Alice to send Bob a validly signed message.

4.4. Suppose that Alice and Bob communicate using the RSA PKC. This means that Alice has a public modulus NA = PAqA, a public encryption exponent eд, and a private decryption exponent dд, where PA and q₁ are primes and eд and dд satisfy eAdA 1 (mod (PA - 1)(A-1)). Similarly, Bob has a public modulus NB = PBqв, a public encryption expo- nent eg, and a private decryption exponent dB. In this situation, Alice can simultaneously encrypt and sign a message in the following way. Alice chooses her plaintext m and computes the usual RSA ciphertext c = mB (mod NB). She next applies a hash function to to her plaintext and uses her private decryption key to compute s = Hash (m)dA (mod N₁). She sends the pair (c, s) to Bob. Bob first decrypts the ciphertext using his private decryption exponent dB, mcdB (mod NB). = He then uses Alice's public encryption exponent eд to verify that Hash (m) SA (mod NA). Explain why verification works, and why it would be difficult for anyone other than Alice to send Bob a validly signed message.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4.4. Suppose that Alice and Bob communicate using the RSA PKC. This means that Alice has a public modulus NA = PAqA, a public encryption exponent eд, and a private decryption exponent dд, where PA and q are primes and eд and dд satisfy едdд = 1 (mod (pa − 1)(qa − 1)). Similarly, Bob has a public modulus NB = PBqB, a public encryption expo- nent eg, and a private decryption exponent dB. In this situation, Alice can simultaneously encrypt and sign a message in the following way. Alice chooses her plaintext m and computes the usual RSA ciphertext c = mB (mod NB). She next applies a hash function to to her plaintext and uses her private decryption key to compute s = Hash (m)dA (mod NA). She sends the pair (c, s) to Bob. Bob first decrypts the ciphertext using his private decryption exponent dB, mcdB (mod NB). He then uses Alice's public encryption exponent eд to verify that Hash (m) SA (mod NA). Explain why verification works, and why it would be difficult for anyone other than Alice to send Bob a validly signed message.
Transcribed Image Text:4.4. Suppose that Alice and Bob communicate using the RSA PKC. This means that Alice has a public modulus NA = PAqA, a public encryption exponent eд, and a private decryption exponent dд, where PA and q are primes and eд and dд satisfy едdд = 1 (mod (pa − 1)(qa − 1)). Similarly, Bob has a public modulus NB = PBqB, a public encryption expo- nent eg, and a private decryption exponent dB. In this situation, Alice can simultaneously encrypt and sign a message in the following way. Alice chooses her plaintext m and computes the usual RSA ciphertext c = mB (mod NB). She next applies a hash function to to her plaintext and uses her private decryption key to compute s = Hash (m)dA (mod NA). She sends the pair (c, s) to Bob. Bob first decrypts the ciphertext using his private decryption exponent dB, mcdB (mod NB). He then uses Alice's public encryption exponent eд to verify that Hash (m) SA (mod NA). Explain why verification works, and why it would be difficult for anyone other than Alice to send Bob a validly signed message.
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