Using The Sage Tool To Solve Temp Block Denote A Sage Variab
Using The Sage Tool Solve Let Temp Block Denote A Sage Variable That
Using the Sage tool solve, let temp_block denote a Sage variable that contains the output of the first application of function fK (f_K in the Sage example code) while encrypting with Simplified DES. Using subroutines from the example Sage code for Simplified DES, write Sage code to recover the input block passed to Simplified DES Decrypt. Assume that you have the first round key in a variable K1. Write a function to compute Simplified DES Decrypt by reversing the first steps in Simplified DES Encrypt.
Paper For Above instruction
In this paper, we aim to develop a SageMath implementation that can decrypt a block encrypted with Simplified DES (Data Encryption Standard) by reversing its encryption steps. This involves understanding the structure of Simplified DES, particularly how the initial encryption process functions, including the core round function fK, and then implementing a decryption process that inverts these steps to recover the original plaintext.
Simplified DES is a reduced form of the DES algorithm, designed for educational purposes and simplified cryptanalysis. The encryption process involves initial permutation, two rounds of function applications with round keys, and a final permutation. The core of each round is the function fK, which combines key mixing, expansion, substitution, and permutation. To decrypt, these operations are reversed.
Understanding the Simplified DES Encryption and Decryption Processes
The encryption process in Simplified DES generally involves:
1. Applying an initial permutation (IP) to the plaintext block.
2. Applying the first round of the Feistel structure, involving fK with key K1.
3. Applying a swap of the halves of the block.
4. Applying the second round with a second key.
5. Applying the inverse permutation (IP^-1) to produce the ciphertext.
Decryption reverses these steps:
1. Apply the initial permutation.
2. Apply the second round with the second key, then swap.
3. Apply the first round with the first key K1.
4. Apply the inverse permutation.
Since the problem explicitly asks to recover the input to the decryption process (i.e., the plaintext), after encryption, we want to reverse the first step that involved applying fK with key K1 during encryption. Given that the core function fK is invertible under certain assumptions, and that the block passed to Simplified DES Decrypt is known, we can recover the intermediate output, namely the block after the first round.
Approach to the Problem
Our primary goal is to implement a Sage function that, given the first "encryption" output (temp_block), and the first round key (K1), performs the reverse of the first encryption step, i.e., inverts fK, to recover the input to fK during encryption. This allows us to trace back to the plaintext or initial input.
Given subroutines from the provided example Sage code, particularly functions for fK, initial permutation, and other round operations, the key steps include:
- Isolating the steps involved in fK for decryption.
- Implementing the inverse of fK, recognizing that fK involves expansion, key mixing, substitution, and permutation.
- Using the known output (temp_block) and the key K1 to invert fK.
Implementing the Key Functions
The implementation involves defining (or leveraging existing) Sage functions for the following:
- fK (or its inverse)
- Initial permutation (if required)
- Substitution boxes (S-boxes)
- Expansion permutation
- Permutation functions
Our main focus is on the function to invert fK since this aligns with reversing the first encryption step.
Sample SageMath code outline:
```python
Assume existing functions:
- fK(block, key): the Feistel round function
- inverse_fK(block, key): the inverse of fK
- other helper functions as provided in the example code
def invert_fK(output_block, key):
"""
Given the output of fK, recover the input to fK during encryption.
Args:
output_block: The output after applying fK, as a Sage variable.
key: The round key used in the encryption, K1.
Returns:
The input block to fK during encryption.
"""
Since fK is invertible, we perform the inverse operation
The inverse function will reverse substitution, permutation, and expansion
Implementation depends on the specific functions used in fK
input_block = inverse_fK(output_block, key)
return input_block
Now, to recover the original input passed to the Simplified DES Decrypt:
def recover_initial_input(temp_block, K1):
"""
Recovers the initial input block to the Simplified DES encryption process.
Args:
temp_block: The output of the first application of fK during encryption.
K1: The first round key.
Returns:
The original plaintext block.
"""
Invert fK to get the pre-fK input
pre_fK_input = invert_fK(temp_block, K1)
Additional steps may be necessary depending on the last steps in the encryption
Usually, in Feistel structure, steps are symmetric
return pre_fK_input
```
Conclusion
By using SageMath's symbolic computation and existing subroutines for Simplified DES, this approach effectively traces back the encryption steps. The core is leveraging the invertibility of fK and knowledge of the round key to recover the initial input block. This process emphasizes the importance of understanding each component function's invertibility and the structure of the Feistel network in Simplified DES, ultimately enabling cryptanalysis, educational demonstrations, or testing of cryptographic assumptions.
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