Note on an additive characterization of quadratic residues modulo p

It is shown that an even partition A∪B of the set R = {1, 2, . . . , p− 1} of positive residues modulo an odd prime p is the partition into quadratic residues and quadratic non-residues if and only if the elements of A and B satisfy certain additive properties, thus providing a purely additive characterization of the set of quadratic residues. 1 Additive properties of quadratic residues An integer a which is not a multiple of a prime p is called a quadratic residue modulo p if the quadratic equation x = a mod p has a solution. If it has no solution then a is called a quadratic non-residue modulo p. The set R = {1, 2, · · · , p−1} of non-zero residues modulo p is evenly partitioned by the quadratic residue character into two sets, A and B, of quadratic residues and quadratic non-residues, respectively. The property of being a quadratic residue or a quadratic non-residue is inherently a multiplicative property, by its definition in terms of field product operations. The paper shows that the set of quadratic residues modulo p can also be characterized strictly in terms of field addition operations. Specifically, it determines the number of ways in which an element c of R can be written as a sum of two elements from A or two elements from B. The answer depends only on whether c is itself an element of A or B. We then show that this property completely determines the sets A and B, providing a purely additive characterization of the set of quadratic residues. Let p be an odd prime, and let QR and QNR stand for quadratic residue and quadratic non-residue, respectively, in the prime field Fp of p elements. Two generating polynomials for the sets of QR and QNR are defined as

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