The security of this encryption algorithm does not only depend on the arbitrariness of the initial state selection.
The final effect of our design in this way is that for each group, the plaintext has L bits, the ciphertext has L bits, and the key has 2L-1 bits. For any known group of plaintext-ciphertext pairs (total 2L bits), all The possible keys (2L-1 bits) can find the initial state S0 and the final state S1 that meet the conditions, and the initial state S0 and the final state S1 have nothing to do with the plaintext. In addition, the encryption of each packet is performed independently.
Therefore, the security of the algorithm is not brought about by the arbitrariness of the initial state selection, but by the algorithm itself, that is to say, you know the arbitrary grouping of plaintext M-ciphertext C pairs, and any key in the key space. K, you can find a correct encryption method (that is, find S0/S1 to M and use K to encrypt the ciphertext C). Therefore, any linear attack or differential attack method is theoretically invalid.

Eagle: A new symmetric encryption algorithm against any linear attacks and differential attacks

Eagle is an encryption algorithm that can resist any linear attacks and differential attacks. Under the encryption mechanism, if any plaintext-ciphertext pair is known, an attacker wants to guess the possible key, he can only do trial-and-error, which satisfy the property of one-way function. In addition, we have also proved theoretically that this kind of problem satisfies the properties of one-way function, so that P!= NP. Since the design mechanism of the algorithm is brand new and can be proved by theory and practical engineering, I look forward to your discussion, thank you.

More details, see https://www.researchgate.net/publication/348992129_Eagle_A_new_symmetric_encryption_algorithm_against_any_linear_attacks_and_differential_attacks_The_existence_of_one-way_function_means_PNP

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The security of this encryption algorithm does not only depend on the arbitrariness of the initial state selection.
The final effect of our design in this way is that for each group, the plaintext has L bits, the ciphertext has L bits, and the key has 2L-1 bits. For any known group of plaintext-ciphertext pairs (total 2L bits), all The possible keys (2L-1 bits) can find the initial state S0 and the final state S1 that meet the conditions, and the initial state S0 and the final state S1 have nothing to do with the plaintext. In addition, the encryption of each packet is performed independently.
Therefore, the security of the algorithm is not brought about by the arbitrariness of the initial state selection, but by the algorithm itself, that is to say, you know the arbitrary grouping of plaintext M-ciphertext C pairs, and any key in the key space. K, you can find a correct encryption method (that is, find S0/S1 to M and use K to encrypt the ciphertext C). Therefore, any linear attack or differential attack method is theoretically invalid.

As in "Theorem 3". we construct an encryption algorithm, as the plaintext-ciphertext is known, the problem of guessing the key can be  transformed to another encryption algorithm,  when only the ciphertext is known, the problem of guessing the key, which satisfies the property of one-way function intuitively.
"Theorem 4" theoretically proves that the above-mentioned problem of guessing the key satisfies the property of one-way function through the calculation of conditional probability. this also proves that the one-way function exists.
The existence of one-way functions can directly prove p!=np, which is a question of common sense.

The encryption algorithm in this paper can be used in all symmetric encryption scheme that require high security.

The features of this algorithm are completely different from those of traditional AES/DES. For the same plaintext and the same key, each encryption will produce a different ciphertext.
For any plaintext-ciphertext pair, any key in the key space is correct. That is, according to any plaintext-ciphertext pair, it is impossible to obtain any characteristics of the key.

I think you misunderstood what i wrote in my paper.
W0 and w1 have an odd number of different bits, which means that they are compared bit by bit, and there are odd numbers of different bits. Or it can be understood that w0 xor w1 has an odd bits of 1.

In addition, I have put the paper on https://www.researchgate.net/publication/348992129_Eagle_A_new_symmetric_encryption_algorithm_against_any_linear_attacks_and_differential_attacks_The_existence_of_one-way_function_means_PNP

Welcome to discuss.

You are right, but there is a problem, it should be S_i-1 XOR (S_i-1 >> 1) = S_i XOR w₁.

If the specific proof is stuck, you can put it away first.

for example.
w0=10010110, w1=01110101
s0=10100010
s0 >> 1 = 01000101
give m=0
s1=w0 xor (s0 xor (s0 >> 1)) = 10010110 xor (10100010 xor 01000101) = 10010110 xor 11100111 = 01110001.
suppose we know s1=01110001, we don't know whether m=0 or m=1, now we guess w=1, we have s0 xor (s0 >> 1) = s1 xor w1 = 00000100.
by the definition of cycle shift, no s0 can be found.

It should be noted that the construction of Eagle's algorithm only relies on uncertainty, not the quality of random numbers. As long as you take any number in the possible value space, using any random number generator will not affect the conclusion.

The quality of the random number generator is concerned with the quality of the random number sequence, which we don’t care about.

if P = NP, then F P = FNP, and so any function that can be computed in polynomial time can be inverted in polynomial time, since there is a simple FNP algorithm that inverts it by nondeterministically enumerating all possible inputs.
However, it is not known whether P = NP implies the existence of one-way functions.

Yes, in this paper, we can replace "randomness" by "arbitrary" without affecting the correctness of the conclusion.

The existence of one-way function means P!=NP

Eagle: A new symmetric encryption algorithm against any linear attacks and differential attacks

Eagle is an encryption algorithm that can resist any linear attacks and differential attacks. Under the encryption mechanism, if any plaintext-ciphertext pair is known, an attacker wants to guess the possible key, he can only do trial-and-error, which satisfy the property of one-way function. In addition, we have also proved theoretically that this kind of problem satisfies the properties of one-way function, so that P!= NP. Since the design mechanism of the algorithm is brand new and can be proved by theory and practical engineering, I look forward to your discussion, thank you.

More details, see https://eaglecrypto.cc/pdf/Eagle.pdf and https://eaglecrypto.cc/pdf/test.pdf

Welcome to email me for details: 20070602094@cqu.edu.cn