A multiplier in an abelian group is an integer power that acts in the same way as a single group element when applied to the elements of a difference set of the group. In this survey paper, the authors provide an overview of progress on the so-called multiplier conjecture. The conjecture states that if p is a prime dividing the order of the difference set but not the order of the group itself, then p is a multiplier. Multipliers are of special interest in cryptography.
The paper begins in section 1 with a description of the history of the conjecture. In section 2, the authors briefly review the number-theoretic and algebraic background needed to understand the theorems and proofs. The authors present in section 3 a slight generalization of McFarland’s result from 1970 on multipliers. In section 4, the authors prove a theorem that guarantees the existence of multipliers of higher order. Such multipliers are needed to verify the conjecture in the cases covered by the results in section 3. In section 5, the authors finally present some computational results that provide more insight into the conjecture. They specifically look at groups of order less than 106.
Overall, this paper, even though it is a survey paper, is not concerned with the application of its results. The authors provide no information about the significance of the conjecture, and they do not include any explanations of applications/implications in cryptography. They instead focus solely on the number theoretical background. This, in my opinion, limits the audience of this paper to mathematicians and researchers working on the same or similar problems. There is also a typographical error in the most basic definition of a multiplier that could lead novice readers astray. Overall, this paper presents an interesting new result but requires a strong background in number theory and only deserves the classification as a survey paper for experts in the field.
