NTRU Modulo p Flaw

Square Roots Modulo p

The algorithm of Tonelli and Shanks for computing square roots modulo a prime number is the most used, and probably the fastest among the known algorithms when averaged over all prime numbers. However, for some particular prime numbers, there are other algorithms which are considerably faster. In this paper we compare the algorithm of Tonelli and Shanks with an algorithm based in quadratic fiel...

Distribution of Residues Modulo p

The distribution of quadratic residues and non-residues modulo p has been of intrigue to the number theorists of the last several decades. Although Gauss’ celebrated Quadratic Reciprocity Law gives a beautiful criterion to decide whether a given number is a quadratic residue modulo p or not, it is still an open problem to find a small upper bound on the least quadratic non-residue mod p as a fu...

Products of Factorials Modulo p

The problem that we investigate in this note is the following: given p, find sufficient conditions that the parameters s and t should satisfy such as to ensure that Ps,t(p) contains the entire Zp. Let ε > 0 be any small number. Throughout this paper, we denote by c1, c2, . . . computable positive constants which are either absolute or depend on ε. From the way we formulated the above problem, w...

Symmetric Pascal matrices modulo p

T =  1 1 1 1 2 1 1 3 3 1 .. . . .  = exp  0 1 0 0 2 0 0 3 0 . . .  with coefficients ti,j = (i j ) . This shows that det(P (n)) = 1 and that P (n) is positive definite for all n ∈ N. It implies furthermore that the characteristic polynomial det(tI(n)−P (n)) = ∑ k=0 αkt k (where I(n) denotes the identity matrix of order n) of P (n) has only positive real roots. The in...

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