Editing Open Problems:53
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{{Header | {{Header | ||
+ | |title=Homomorphic hash functions | ||
|source=dortmund12 | |source=dortmund12 | ||
|who=Ely Porat | |who=Ely Porat | ||
}} | }} | ||
β | '''Question | + | '''Question''': to construct a hash function |
$ | $ | ||
h:\F_p^n \to \F_p^m | h:\F_p^n \to \F_p^m | ||
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* for any $u,v$, we have $\Pr_h[h(u)=h(v)]=\frac{c}{p^m}$ for some constant $c$ independent of $n,m$. | * for any $u,v$, we have $\Pr_h[h(u)=h(v)]=\frac{c}{p^m}$ for some constant $c$ independent of $n,m$. | ||
β | One solution is by considering a random linear function, given by the matrix $M$. Then we have that $\Pr_M[Mu=Mv]=\Pr_M[M(u-v)=0]=1/p^m$. This function would require $O(nm\log p)$ random bits, and computing $h$ takes $O(nm)$ time. We would like more efficient solutions. Ely and coauthors claim a solution with $O((n+m)\log p)$ bits, and $O((n+m)\log (n+m))$ time. If one considers Reed-Salomon codes, it seems that they would give worse bound on second property (probability of collision). | + | One solution is by considering a random linear function, given by the matrix $M$. Then we have that $\Pr_M[Mu=Mv]=\Pr_M[M(u-v)=0]=1/p^m$. This function would require $O(nm\log p)$ random bits, and computing $h$ takes $O(nm)$ time. We would like more efficient solutions. |
+ | |||
+ | Ely and coauthors claim a solution with $O((n+m)\log p)$ bits, and $O((n+m)\log (n+m))$ time. | ||
+ | |||
+ | If one considers Reed-Salomon codes, it seems that they would give worse bound on second property (probability of collision). |