The best methods for storing seed potatoes begin before harvest by cutting or knocking down the vines a week or two before digging the spuds. This gets the skins to toughen up for the handling that comes with harvest and storage. After harvest, an intentional skin healing, curing or "suberization" period of a couple weeks keeping at them 55-65F is helpful before dropping the temperature to 35-39F for storing for next seasons planting. Some people choose the golf ball size potatoes for seed so that cuts are not necessary when planting in the Spring.
See the two UAF Extension publications "Growing Potatoes in the Alaska Garden" (HGA-00027) and "Small-Scale Potato Storage Management" (HGA-01024) for more info.