## Internet Mathematics

- Internet Math.
- Volume 1, Number 2 (2003), 177-192.

### Guessing Secrets with Inner Product Questions

Fan Chung, Ronald Graham, and Linyuan Lu

#### Abstract

We suppose we are given some fixed (but unknown) subset *X* of a set $\Omega = {\mathbb F}_2^n$,
where ${\mathbb F}_2$ denotes the field of two elements. Our goal is to learn as much as possible about the elements of *X* by asking certain binary questions. Each "question" *Q* is just some element of $\Omega$, and the "answer" to *Q* is just the inner product $Q \cdot x \in {\mathbb F}_2$ for *some* $x \in X$. However, the choice of *x* is made by a truthful (but possibly malevolent) adversary **A**, whom we may assume is trying to choose answers so as to yield as little
information as possible about *X*. In this note, we investigate several aspects of this problem. In particular, we are interested in extracting as much information as possible about *X* from **A**'s answers. Although **A** can prevent us from learning the identity of any particular element of *X*, with appropriate questions we can still learn quite a bit about *X*. We determine the maximum amount of information that can be recovered under these assumptions and describe explicit sets of questions for achieving this goal. For the case that $|X|=2$, we give an $O(n^3)$ algorithm for recovering the desired information. On the other hand, when $|X| \geq 3$, we show that no polynomial-time algorithm can exist for producing a secret set consistent with the answers given, unless $P = NP$.

#### Article information

**Source**

Internet Math., Volume 1, Number 2 (2003), 177-192.

**Dates**

First available in Project Euclid: 7 July 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.im/1089229507

**Mathematical Reviews number (MathSciNet)**

MR2077224

**Zentralblatt MATH identifier**

1058.94006

#### Citation

Chung, Fan; Graham, Ronald; Lu, Linyuan. Guessing Secrets with Inner Product Questions. Internet Math. 1 (2003), no. 2, 177--192. https://projecteuclid.org/euclid.im/1089229507