Short integer solution problem

Short integer solution (SIS) and ring-SIS problems are two average-case problems that are used in lattice-based cryptography constructions. Lattice-based cryptography began in 1996 from a seminal work by Miklós Ajtai^[1] who presented a family of one-way functions based on SIS problem. He showed that it is secure in an average case if the shortest vector problem ${\displaystyle \mathrm {SVP} _{\gamma }}$ (where ${\displaystyle \gamma =n^{c}}$ for some constant ${\displaystyle c>0}$) is hard in a worst-case scenario.

Average case problems are the problems that are hard to be solved for some randomly selected instances. For cryptography applications, worst case complexity is not sufficient, and we need to guarantee cryptographic construction are hard based on average case complexity.