A primality test using cyclotomic extensions

Abstract

The cyclotomic polynomial Φ_s(x) (where s is an integer >1) is the irreducible polynomial over ℚ, having the primitive s-th roots of unity as zeroes. If $\mathbb{K}$ is the field ℚ or $\mathbb{F}_p$, with p a prime, an s-th cyclotomic extension of $\mathbb{K}$ is the splitting field of Φ_s(x) over $\mathbb{K}$. Every cyclic (actually: every abelian) extension of $\mathbb{K}$ is included in some cyclotomic one. We present in §2 a procedure for constructing cyclic and cyclotomic extensions of fields. Cyclic fields are used in [BS] in the context of a factoring algorithm. For $\mathbb{K} = \mathbb{F}_p$, this procedure can be used to produce irreducible polynomials of given degree over $\mathbb{F}_p$.

H.W.Lenstra, Jr. has extended in [Le2] the concept of cyclotomic extensions to rings ℤ/(n ℤ), with n>1 an integer, and showed that existence of such extensions of degree s>√n implies a drastical constraint upon possible prime factors of n. He proposes a primality test based upon factoring Φ_s(x) over ℤ/(n ℤ), using the Berlekamp algorithm. Using our algorithm for constructing cyclotomic extensions over a finite field and some pseudoprime test involving Jacobi sums, we improve in §5,6 Lentra's approach to proving existence of an s-th cyclotomic extension of ℤ/(n ℤ) in polynomial time, when s>√n and ord_s(n) = O(log(n)^{c.logloglog(n)}). This leads to a new primality test, which is an algorithmical realisation of the sketches in [Le2]. The intimate connection with the Adleman test ([APR],[Le1],[CoLe]) becomes evident by the very similar algebraic techniques used in that test and in ours. The new algorithm is comparable to [CoLe] in assymptotic runtime and capacity to prove primality of a test number; it is slightly superior by the fact that, (a) in the most time consuming steps of both algorithms, the set of operations required by CE is a subset of the set of operations required by Jacobi-sum test as described in [CoLe] and (b) the new algorithm provides a proof of primality in all cases the test [CoLe] does so, and also in some further cases.