Comparing stronger vs. weaker assumptions

Question

If I have to argue that an assumption A is stronger than B, should I argue this way:-

An adversary breaking the security of A does not translate to breaking the security of B. But, if an adversary breaks the security of B, it can break A too.

Right? or, there are other ways to compare stonger vs. weaker assumptions.

Answer 1

Yes you are right. Essentially when we say $A$ is a stronger assumption than $B$, what we mean is that assumption $A$ implies assumption $B$. $A$ implies $B$ tells us that:

• if $A$ is true, then $B$ must be true.

$A$ implies $B$ is also equivalent to $\neg B$ implies $\neg A$, thus

• if $B$ is false, then $A$ must be false.

In cryptography, assumptions often are related to the hardness of some problems. In this context, stronger assumptions are "easier" problems that can be reduced to weaker assumptions that are "harder"problems (solving the harder problems means the easier ones are also solvable). For example, say we have the following two assumptions:

• $A$: factoring an integer $n$ is difficult;
• $B$: factoring an integer $n$ with only large prime factors is difficult.

Clearly, if $A$ is true, then $B$ is true; if $B$ is false, $A$ is false. If we can solve $B$ then we can use the algorithm that solves $B$ to solve $A$.