We give a sieving algorithm for finding pairs of consecutive smooth numbers that utilizes solutions to the Prouhet-Tarry-Escott (PTE) problem. Any such solution induces two degree-\\(n\\) polynomials, \\(a(x)\\) and \\(b(x)\\), that differ by a constant integer \\(C\\) and completely split into linear factors in \\(\\Z[x]\\). It follows that for any \\(\\ell \\in \\Z\\) such that \\(a(\\ell) \\equiv b(\\ell) \\equiv 0 \\bmod{C}\\), the two integers \\(a(\\ell)\/C\\) and \\(b(\\ell)\/C\\) differ by 1 and necessarily contain \\(n\\) factors of roughly the same size. For a fixed smoothness bound \\(B\\), restricting the search to pairs of integers that are parameterized in this way increases the probability that they are \\(B\\)-smooth. Our algorithm combines a simple sieve with parametrizations given by a collection of solutions to the PTE problem.<\/p>\n\n\n\n
The motivation for finding large twin smooth<\/em> integers lies in their application to compact isogeny-based post-quantum protocols. The recent key exchange scheme B-SIDH and the recent digital signature scheme SQISign both require large primes that lie between two smooth integers; finding such a prime can be seen as a special case of finding twin smooth integers under the additional stipulation that their sum is a prime \\(p\\).<\/p>\n\n\n\n When searching for cryptographic parameters with \\(2^{240} \\leq p <2^{256}\\), an implementation of our sieve found primes \\(p\\) where \\(p+1\\) and \\(p-1\\) are $2^{15}$-smooth; the smoothest prior parameters had a similar sized prime for which \\(p-1\\) and \\(p+1\\) were $2^{19}$-smooth. In targeting higher security levels, our sieve found a 376-bit prime lying between two $2^{21}$-smooth integers, a 384-bit prime lying between two $2^{22}$-smooth integers, and a 512-bit prime lying between two $2^{29}$-smooth integers. Our analysis shows that using previously known methods to find high-security instances subject to these smoothness bounds is computationally infeasible.<\/p>\n","protected":false},"excerpt":{"rendered":" We give a sieving algorithm for finding pairs of consecutive smooth numbers that utilizes solutions to the Prouhet-Tarry-Escott (PTE) problem. Any such solution induces two degree- polynomials, and , that differ by a constant integer and completely split into linear factors in . 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