Digital signatures, continued.Consider the signature scheme of Exercise $\mathbf{1}\mathbf{.45}$.

(a) Signing involves decryption, and is therefore risky. Show that if Bob agrees to sign anything he is asked to, Eve can take advantage of this and decrypt any message sent by Alice to Bob.

(b) Suppose that Bob is more careful, and refuses to sign messages if their signatures look suspiciously like text. (We assume that a randomly chosen message$\mathbf{-}$that is, a random number in the range$\mathbf{\{}\mathbf{1}\mathbf{,}\mathbf{.}\mathbf{..}\mathbf{,}\mathbf{N}\mathbf{-}\mathbf{1}\mathbf{\}}$ is very unlikely to look like text.) Describe a way in which Eve can nevertheless still decrypt messages from Alice to Bob, by getting Bob to sign messages whose signatures look random.

The public-key cryptosystem allows sending messages between Alice and Bob.The third-party Eve tries to access the message using some calculations on the hash function.The digital locking of the keys provides security to the communication.The security is implemented using RSA digital signature methods.

(a)

Using the RSA algorithm, Alice sends the encrypted message$E=M^e\mathrm{mod}N$to Bob.

During decryption, when Eve tries to intercept the message, then Alice asks Bob that to just sign in using his private key to get the message$M={\left(M^e\right)}^d\mathrm{mod}N$.

This process of RSA encryption is ensured by a digital signature certificate.The Certificate Authority (CA) prevents Eve from accessing the key pair of Alice.The CA claims that it is Bob’s public key.

Suppose, Eve tries to send a message with the sign of Bob, Alice checks the validated public key of Bob and compares the hash result.

Then it fails the decryption. Thus, it will not work for Eve.

Therefore, the encryption and decryption of the message are secured by using a digital signature.

(b).

The correctness of the RSA algorithm is used for implementing the decryption procedure.By using the procedure, the access of Bob’s key by Eve does not work.A locking key pair is generated.So that, Eve cannot tamper with the message.

Decryption procedure:

The decryption procedure for the message that Eve does not get access to the key by using a random number is as follows.

• Choose two random prime numbers$r,s$.
• Calculate$n=r\times s$.
• Calculate$\varnothing \left(n\right)=(r-1)\times (s-1)$.
• Choose a numbersuch as$1<e<\varnothing \left(n\right)$.
• The number $e$is co-prime to$\varnothing \left(n\right)$.
• Calculate$\gcd (e,\varnothing \left(n\right))=1$.
• Calculate the key$d$such that$d.e\equiv 1\mathrm{mod}\varnothing \left(n\right)$.
• Public key is taken as$e$and the private key is taken as$d$.

By using the above procedure, the public and private key pairs are generated.The public key of Alice is known to Bob and vice versa.

Thus, Eve cannot access the message from Alice to Bob.

Therefore, the procedure for decryption is developed in such a way that Eve cannot access the message of Alice and Bob.