Invertible calculation's non-invertibility

In cryptosystem theory, it is well-known that a logical mapping that returns the same value that was used as its argument can be inverted with a zero failure probability in linear time. In this paper, however, I show that such a mapping, while trivially assigning a logical state to itself, comes up against an impassable entropy wall such that its computational path is undone inside the thresholds of the physical world. Present cryptology-related theory is heavily based on an unproven one-wayness proposition of computational paths (1-5). This mathematical conjecture holds that there must be a one-to-one correspondence for which the calculation in one direction is easy, while reconstructing the input state from the output state is hard – "easy" and "hard" are to be understood in the sense of time-complexity (3,4,6,7). More specifically, this computational hardness proposition essentially requires the existence of an invertible function that is non-invertible (6,7). In the last few years, the idea has arisen that a proof of computational hardness is linked to physical constraints rather than purely mathematical limitations (8-12). Taking this new perspective into account, one could preliminarily infer that all previous efforts to prove the existence of a one-way function have been doomed to failure because we have relied on a mathematical approach that does not correspond to physical reality. Based on this new perspective, I present an alternative scenario that connects the invertibility condition of a particular bijective mapping to the thermodynamic bounds of computation. In what follows, I argue that while a logical operation maps its input state to the output state unchanged can be computed efficiently, the computation of its inverse cannot be computed efficiently because undoing the computational path of this sort of correspondence violates a most important principle of nature: the second law of thermodynamics. To give reasons for accepting this claim as evident, I explore the entropy bounds of a full computation's cycle by running backward and forward the well-known Maxwell’s demon gedankenexperiment (13-15), whose feedback control engine manipulates a measured system based on its thermal fluctuations into its logical memory to restore its standard state (16). Let us then distinguish a couple of logical structures in our back-and-forth mechanism, namely, a binary memory and a measured system, whose logical states are ruled by a Controlled-NOT operator (C-NOT gate), wherein the memory is the target-bit. Assuming here the widely accepted Bennett’s algorithm of Maxwell’s demon (14,17), such a Controlled-NOT operation should so be conceived as to be mathematically embodied in an information heat engine so that both the binary memory and the measured system should be designed as a heat reservoir and an adiabatic box, respectively. In this way, the demon's memory records the thermal fluctuations from the measured system. To measure and restore the memory’s standard state, Bennett’s algorithm, fundamentally, utilizes two non-commutative stages (14,16,17). Initially, a removable partition is inserted (with no thermodynamic work) into the middle of the adiabatic box resulting in the splitting of the memory into a twofold states. Before the insertion, as shown in Fig. 1(a) and (b), the C-NOT gate operator stores the memory’s state in the target-bit “0” with a probability equal to unity, while the control-bit is in a “0” or “1”. After the insertion of the partition, as shown in Fig. 1(c), the logical state of the measuring system becomes perfectly correlated to the logical state of the measured system. This operation does not result in any heat exchange with the measured system and corresponds to the target-bit being flipped whether the control-bit is “1” in the C-NOT gate. In other words, the measurement (acquisition of information from Bennett's conception) induces a randomization of the memory's state with a bit equally likely to be “0” or “1” (this will become quite clear below). In the next stage, a logical merging of two states into one should occur while performing a loop's closing. If we merge data from a symmetric double-well memory’s state, then there must be a change in some other macroscopic variable of the ensemble. Liouville's theorem requires that this change be a volume-preserving operation, where the state space should remain invariant under the transformation, provided that the region available to the logical degrees of freedom is reduced by a factor of two and that the region available to the non-information degrees of freedom is doubled