An Example to See the Advantages of ChatGPT V.S. NewBing

Give me an explanation of Fk(r)F_k(r)Fk​(r) in the book of Introduction to Modern Cryptography. (Q)

This is a text I got from the book named Introduction to Modern Cryptography. I want to know the fucntion of $<>$ in above text.

Construction 3.28 Let $F$ be a pseudorandom function. Define a fixed-length, private-key encryption scheme for messages of length $n$ as follows:

• Gen: on input 1n1^n1n, choose uniform k∈{0,1}nk \\in \\{0, 1\\}^nk∈{0,1}n and output it.
• Enc: on input a key k∈{0,1}nk \\in \\{0, 1\\}^nk∈{0,1}n and a message m∈{0,1}nm \\in \\{0, 1\\}^nm∈{0,1}n, choose uniform r∈{0,1}nr \\in \\{0, 1\\}^nr∈{0,1}n and output the ciphertext c:=<r,Fk(r)⊕m>c:=<r, F_k(r) \\oplus m>c:=<r,Fk​(r)⊕m>
• Dec: on input a key k∈{0,1}nk \\in \\{0, 1\\}^nk∈{0,1}n and a ciphertext $c=<r,s>$, output the message m:=Fk(r)⊕sm := F_k(r) \\oplus sm:=Fk​(r)⊕s.

1. The answer from GPT
   In the construction you provided, the notation <r,Fk(r)⊕m><r, F_k(r) \\oplus m><r,Fk​(r)⊕m> is used to represent the concatenation of the two bit strings $r$ and Fk(r)⊕mF_k(r) \\oplus mFk​(r)⊕m.
   The angle brackets $<>$ are often used to denote concatenation of strings or tuples in cryptography and computer science. In this case, the ciphertext is composed of two parts: the random value $r$ and the XOR of the key-dependent value Fk(r)F_k(r)Fk​(r) with the plaintext message $m$. These two parts are concatenated together using the angle bracket notation to form the complete ciphertext.
2. The answer from NewBing
   In the context of the text you provided from the book “Introduction to Modern Cryptography,” the angle brackets $<>$ are used to denote an ordered pair or tuple. In this case, the ciphertext $c$ is an ordered pair consisting of two elements: $r$ and Fk(r)⊕mF_k(r) \\oplus mFk​(r)⊕m. The first element of the pair is $r$, and the second element is the result of the bitwise exclusive or (XOR) operation between the output of the keyed function Fk(r)F_k(r)Fk​(r) and the message $m$. Similarly, when decrypting, the ciphertext $c$ is split into its two components, $r$ and $s$, where $s$ represents the second element of the pair.

In addition, I want to know how to understand the fucntion of Fk(r)F_k(r)Fk​(r) in above context.

1. The Answer from GPT

   • In the context of the construction you provided, Fk(r)F_k(r)Fk​(r) refers to the output of a pseudorandom function $F$ evaluated at the point $r$, using the secret key $k$ as input.

   • The purpose of using Fk(r)F_k(r)Fk​(r) in the encryption process is to introduce randomness into the ciphertext, making it more difficult for an attacker to learn information about the plaintext message from the ciphertext. The value Fk(r)F_k(r)Fk​(r) is used to “mask” the plaintext message $m$ by XORing it with Fk(r)F_k(r)Fk​(r), which is itself a pseudorandom value that is indistinguishable from a truly random value.

   • The security of the encryption scheme depends on the assumption that $F$ is a secure pseudorandom function, meaning that it is indistinguishable from a truly random function. If this assumption holds, then an attacker who does not know the secret key $k$ will not be able to learn any information about the plaintext message from the ciphertext, even if they have access to many ciphertexts encrypted under the same key.

2. The Answer from NewBing
   • In the context of the text you provided from the book “Introduction to Modern Cryptography,” Fk(r)F_k(r)Fk​(r) represents the output of a pseudorandom function $F$ when given the key $k$ and input $r$. A pseudorandom function is a deterministic function that takes a secret key and an input, and produces an output that is indistinguishable from a truly random function to any efficient adversary who does not know the key. In this case, the pseudorandom function is used to generate a pseudorandom output Fk(r)F_k(r)Fk​(r), which is then XORed with the message $m$ to produce the second element of the ciphertext. During decryption, the same pseudorandom output Fk(r)F_k(r)Fk​(r) is generated again using the same key and input, and then XORed with the second element of the ciphertext to recover the original message.

Give me an example to see the specific calculation process for Fk(r)F_k(r)Fk​(r).(Q)