{'fraction_non_alphanumeric': 0.023107569721115537, 'fraction_numerical': 0.0, 'mean_word_length': 5.248756218905473, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Leveled Fully Homomorphic Encryption Schemes with Hensel Codes

We propose the use of Hensel codes (a mathematical tool lifted from the theory of p-adic numbers) as an alternative way to construct fully homomorphic encryption (FHE) schemes that rely on the hardness of some instance of the approximate common divisor (AGCD) problem. We provide a self-contained introduction to Hensel codes which covers all the properties of interest for this work. Two constructions are presented: a private-key leveled FHE scheme and a public-key leveled FHE scheme. The public-key scheme is obtained via minor modifications to the private-key scheme in which we explore asymmetric properties of Hensel codes. The efficiency and security (under an AGCD variant) of the public-key scheme are discussed in detail. Our constructions take messages from large specialized subsets of the rational numbers that admit fractional numerical inputs and associated computations for virtually any real-world application. Further, our results can be seen as a natural unification of error-free computation (computation free of rounding errors over rational numbers) and homomorphic encryption. Experimental results indicate the scheme is practical for a large variety of applications.