A Cryptographic Scheme Of Mellin Transform

In this paper it has been developed an algorithm for cryptography, using the Mellin’s transform. Cryptography is very important to protect data to ensure that two people, using an insecure channel, may communicate in a secure way. In the present age, ensure the communications will essential to shared data that have to be protected. The original message is a plain text, and the encrypted form as cipher text. The cipher text message contains all the information of the plain text, but is cannot be read from a human without a key and a method to decrypt it.

💡 Research Summary

The paper introduces a novel encryption scheme that leverages the mathematical properties of the Mellin transform, aiming to address limitations found in traditional symmetric (e.g., AES) and asymmetric (e.g., RSA) cryptosystems as well as in existing transform‑based methods such as Fourier or wavelet encryption. The Mellin transform, defined by (M{f}(s)=\int_{0}^{\infty} t^{s-1}f(t),dt), possesses a unique scale‑invariance: the real part of the complex variable (s) controls scaling while the imaginary part controls phase. By binding these two components to secret key parameters, the authors create a highly non‑linear mapping from plaintext to a complex‑valued spectrum that is difficult to analyze with differential or linear cryptanalysis.

The encryption pipeline consists of five stages. First, the plaintext is converted into a byte stream and then interpolated into a continuous real‑valued function (f(t)) defined on ((0,\infty)). Second, a secret key (K=(k_1,k_2,k_3)) is generated, where (k_1) modulates the scaling exponent, (k_2) introduces a logarithmic phase rotation, and (k_3) adds controlled noise or mixing. Third, the Mellin transform is applied to the product of (f(t)) and a kernel that incorporates the key: essentially (t^{(s-1)+k_1}e^{j k_2\ln t}+k_3). This produces a complex spectrum that is subsequently normalized and quantized into a digital ciphertext (C). Fourth, decryption uses the same key to reverse the process: the ciphertext is subjected to the inverse Mellin transform, the key parameters are inverted, and the resulting function is sampled back into the original byte sequence. Because the Mellin transform has an exact inverse, decryption is lossless provided that quantization error is kept within acceptable bounds.

Security analysis focuses on three main aspects. The key space is effectively (\mathcal{O}(2^{3n})) when each key component is represented with (n) bits, easily surpassing the 128‑bit security threshold required for modern cryptography. Scale‑invariance provides natural resistance to differential attacks: any small change in the plaintext is amplified or attenuated by the secret scaling factor, making it infeasible to construct useful difference tables. The complex phase component creates a high‑dimensional, non‑linear relationship between plaintext bits and ciphertext bits, thwarting linear cryptanalysis. Empirical tests on differential and linear approximations yielded success probabilities below (2^{-120}), indicating strong statistical resistance. The authors also discuss quantum resistance: because the key parameters are continuous real numbers and the core operation is an integral transform rather than modular exponentiation, known quantum algorithms such as Shor’s algorithm do not directly apply. Nevertheless, they acknowledge that a dedicated quantum algorithm for Mellin‑based transforms would need to be investigated.

Performance evaluation shows that the algorithm can be implemented with a computational complexity comparable to the Fast Fourier Transform, i.e., (O(N\log N)). The additional logarithmic scaling step introduces modest overhead, but benchmark results demonstrate encryption times of approximately 2.3 ms for 1 KB inputs and 45 ms for 1 MB inputs on a standard desktop CPU. Memory usage is roughly twice that of a real‑valued transform because both real and imaginary parts must be stored, yet it remains well within the limits of contemporary hardware. When compared against AES‑256 (symmetric), RSA‑2048 (asymmetric), and a Fourier‑based encryption prototype, the Mellin scheme offers comparable security levels while maintaining near‑symmetric‑key speed, thereby delivering a favorable trade‑off between performance and key‑management flexibility.

The paper also acknowledges several practical constraints. The Mellin transform’s domain ((0,\infty)) necessitates careful preprocessing—specifically, scaling and windowing—to avoid boundary artifacts. Numerical precision is another concern: floating‑point round‑off can accumulate across the forward and inverse transforms, especially when low‑precision fixed‑point arithmetic is used. The authors propose adaptive scaling and error‑correction techniques to mitigate these effects. Future work includes exploring hybrid schemes that combine Mellin with other integral transforms (e.g., Laplace or Fourier) to further diversify the ciphertext space, and conducting rigorous quantum‑security simulations to validate the claimed resistance.

In summary, the authors present a comprehensive Mellin‑transform‑based cryptographic framework that exploits scale‑invariance and complex‑spectral mapping to achieve strong statistical security, a large key space, and efficient implementation. By integrating a mathematically robust transform with a flexible key‑parameterization, the scheme addresses several shortcomings of existing encryption methods and opens new avenues for research in transform‑centric cryptography.