This paper investigates, through an exhaustive method, the statistical properties of the self-power mapon the multiplicative group of finite fields of characteristic 2, where p takes the Mersenne prime exponents 5,7,13,17,19,315,7,13,17,19,31 and the group order is a Mersenne prime. For each group, we traverse all nonzero elements x compute H(x), and analyze the preimage distribution of the output values. The results show that, for all tested Mersenne prime groups, the number of preimages precisely follows a Poisson distribution Poi(1), the size of the image set is exactly, and the collision probability is 1/n, perfectly consistent with the ideal random mapping model. The chi-square goodness-of-fit test indicates no significant deviation between observation and theory (). In particular, the exhaustive computation for p=31 (scale ) provides the first decisive evidence on a large domain. Avalanche effect tests further confirm the local randomness of this map: for parameters p=19,31,127,521,1279 and others, the distribution of the number of flipped bits exactly matches the binomial distribution Bin(p,1/2). Based on this discovery, the author proposes several cryptographic applications, including provably secure hash functions, message authentication codes, symmetric ciphers, and stream cipher constructions, and analyzes their security under classical and quantum computing. This study provides the first complete empirical foundation for the randomness theory of power maps over finite fields and introduces new, provably secure cryptographic primitives designed to withstand quantum computing attacks cryptography.
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