P vs NP problem is the most important unresolved problem in the field of computational complexity. Its impact has penetrated into all aspects of algorithm design, especially in the field of cryptography. The security of cryptographic algorithms based on short keys depends on whether P is equal to NP. In fact, the security requirements for cryptographic keys are much stricter than those for P$\neq$NP, the security of the key must ensure not only a sufficiently high computational complexity to crack it, but also consider the security of each bit of the key, while fully avoiding the effectiveness of various attack methods. In this paper, we innovatively propose a new encoding mechanism and develop a novel block symmetric encryption algorithm, whose encryption and decryption can be completed in linear time. For the attacker, in the case where only the plaintext-ciphertext correspondence is known, the problem of cracking the key is equivalent to solving a system of equations which contains at least one free variable that cannot be eliminated, and the number of possible values for each variable is exponentially to the length of the key. To solve this system of equations, it is necessary to exhaustively search for at least one variable, thus proving that the computational complexity of cracking the key is exponential. So the decryption is a one-way function, and according to "the existence of one-way function means P$\neq$NP", thus solving the unsolved problem of P vs NP. In addition, this paper delves into the underlying mathematical laws of this new encoding mechanism, and develops a right multiplication operation to binary. Based on this right multiplication operation, we further constructed a nonlinear operation and designed another block symmetric encryption algorithm that is resistant to all forms of linear and differential attacks.

翻译：P与NP问题是计算复杂性领域最重要的未解难题，其影响已渗透至算法设计的各个方面，尤其在密码学领域。基于短密钥的密码算法的安全性取决于P是否等于NP。实际上，对密码密钥的安全性要求远比P$\neq$NP更为严格：密钥不仅需要保证足够高的破解计算复杂度，还必须考虑密钥每一位的安全性，同时完全规避各类攻击方法的有效性。本文创新性地提出一种新型编码机制，并构建了一种新颖的分组对称加密算法，其加密与解密过程均可在线性时间内完成。对于攻击者而言，在仅知明文-密文对应关系的情况下，破解密钥的问题等价于求解一个至少包含一个无法消去的自由变量的方程组，且每个变量的可能取值数量与密钥长度呈指数关系。求解该方程组必须对至少一个变量进行穷举搜索，从而证明破解密钥的计算复杂度是指数级的。因此解密过程构成单向函数，依据"单向函数的存在意味着P$\neq$NP"，由此解决了P与NP这一悬而未决的难题。此外，本文深入探究了该新型编码机制的底层数学规律，并针对二进制数构建了右乘运算。基于此右乘运算，我们进一步构造了非线性操作，设计了另一种能抵抗所有形式线性攻击与差分攻击的分组对称加密算法。