Counterexamples to Hardness Amplification beyond Negligible

If we have a problem that is mildly hard, can we create a problem that is significantly harder? A natural approach to

hardness amplification

is the “direct product”; instead of asking an attacker to solve a single instance of a problem, we ask the attacker to solve several independently generated ones. Interestingly, proving that the direct product amplifies hardness is often highly non-trivial, and in some cases may be false. For example, it is known that the direct product (i.e. “parallel repetition”) of general interactive games may not amplify hardness at all. On the other hand, positive results show that the direct product does amplify hardness for many basic primitives such as one-way functions, weakly-verifiable puzzles, and signatures.

Even when positive direct product theorems are shown to hold for some primitive, the parameters are surprisingly weaker than what we may have expected. For example, if we start with a weak one-way function that no poly-time attacker can break with probability

$> \frac{1}{2}$

, then the direct product provably amplifies hardness to

some negligible

probability. Naturally, we would expect that we can amplify hardness exponentially, all the way to 2

− 

n

probability, or at least to some fixed/known negligible such as

n

− log

n

in the security parameter

n

, just by taking sufficiently many instances of the weak primitive. Although it is known that such parameters cannot be proven via black-box reductions, they may seem like reasonable conjectures, and, to the best of our knowledge, are widely believed to hold. In fact, a conjecture along these lines was introduced in a survey of Goldreich, Nisan and Wigderson (ECCC ’95). In this work, we show that such conjectures are

false

by providing simple but surprising counterexamples. In particular, we construct weakly secure

signatures

and

one-way functions

, for which standard hardness amplification results are known to hold, but for which hardness does not amplify

beyond just negligible

. That is, for

any negligible

function

$\ensuremath{\varepsilon} (n)$

, we instantiate these primitives so that the direct product can always be broken with probability

$\ensuremath{\varepsilon} (n)$

,

no matter how many copies we take

.