Using the Chinese remainder theorem, we can prove that implementing as
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By an ingenious use of Chinese remainder theorem, we can define constructively such a recursive function (using simple number-theoretical functions, all of which can be defined in a total recursive way) fulfilling the "specifications" given above.
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Using the fundamental theorem of arithmetic is a straightforward way, but there are also more economic approaches, e.g. using pairing function combined with Chinese remainder theorem in a sophisticated way.
Then, for any sets U and V, by the Chinese remainder theorem, the numbers that are quadratic resides modulo every prime in U and nonresidues modulo every prime in V form a periodic sequence, so by Dirichlet's theorem on primes in arithmetic progressions this number-theoretic graph has the extension property.
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In 2003, Boneh and Brumley demonstrated a practical network-based timing attack on SSL-enabled web servers, based on a different vulnerability having to do with the use of RSA with Chinese remainder theorem optimizations.