Cryptographic systems and non-deterministic random number generators based on quantum systems

A system for providing a random bit string for use in one or more cryptographic applications, the system comprising:

• an electronic processor that executes specific computer-executable instructions stored in a non-transitory memory to provide a two-source randomness extractor and a seeded randomness extractor, wherein the specific computer-executable instructions when executed cause the electronic processor to at least:
• receive, by the two-source randomness extractor, from a first source of randomness a first input comprising a first random bit string having an input length, represented by n, a first min-entropy, and a first min-entropy rate;
• receive, by the two-source randomness extractor, from a second source of randomness a second input comprising a second random bit string having a second min-entropy and a second min-entropy rate;
• generate and output, by the two-source randomness extractor, a third random bit string derived from the first and second random bit strings, the third random bit string having a third min-entropy rate which has been amplified to be greater than at least the first min-entropy rate;
• receive, by the seeded randomness extractor, from the second source of randomness, a fourth random bit string having a fourth min-entropy, wherein the fourth min-entropy is less than a third min-entropy of the third random bit string;
• receive, by the seeded randomness extractor, the third random bit string as a seed;
• and
• generate, by the seeded randomness extractor, a fifth random bit string using the third and fourth random bit strings;
• wherein a length of the third random bit string, represented by m, is asymptotically proportional to a sum of the first and second min-entropies minus the input length (n);
• and wherein the two-source randomness extractor has a complexity of order equal to or less than n×log (n);
• and
• a key generation system that receives the fifth random bit string and uses the fifth random bit string to generate a cryptographic key used in encryption or decryption of data.