Identification and signatures based on NP-hard problems of indefinite quadratic forms

Identification and signatures based on NP-hard problems of indefinite quadratic forms

We prove NP-hardness of equivalence and representation problems of quadratic forms under probabilistic reductions, in particular for indefinite, ternary quadratic forms with integer coefficients. We present identifications and signatures based on these hard problems. The bit complexity of signature generation and verification is quadratic using integers of bit length 150.

On successive minima of indefinite quadratic forms

On Indefinite Binary Quadratic Forms and Quadratic Ideals

We consider some properties of indefinite binary quadratic forms F (x, y) = ax +bxy−y of discriminant ∆ = b +4a, and quadratic ideals I = [a, b−√∆ ]. AMS Mathematics Subject Classification (2000): 11E04, 11E12, 11E16

Quadratic Irrationals, Quadratic Ideals and Indefinite Quadratic Forms II

Let D = 1 be a positive non-square integer and let δ = √ D or 1+ √ D 2 be a real quadratic irrational with trace t = δ + δ and norm n = δδ. Let γ = P+δ Q be a quadratic irrational for positive integers P and Q. Given a quadratic irrational γ, there exist a quadratic ideal Iγ = [Q, δ + P ] and an indefinite quadratic form Fγ(x, y) = Q(x−γy)(x−γy) of discriminant Δ = t − 4n. In the first section,...

Bailey Pairs and Indefinite Quadratic Forms

We construct classes of Bailey pairs where the exponent of q in αn is an indefinite quadratic form. As an application we obtain families of q-hypergeometric mock theta multisums.