as noted above, the polynomial φ(x) which defines the ring of polynomials used will be x + 1. finally, a(x) will be a randomly chosen and fixed polynomial with coefficients from the set { -(q-1)/2 to (q-1)/2 }. the polynomial a(x) should be chosen in a "nothing up my sleeve" manner such as one-way hashing the output of a true noise random number generator (trng) or using the digital expansion of well known mathematical constans such as pi or e. all signers and verifiers of signatures will know n, q, b, k, φ(x), a(x) and β = b-k.