storing a private key in k key fragments that include k−1 key segments and a remainder key fragment, where k is greater than 2, wherein the k−1 key segments are generated as random key segments, and wherein {HEAD ~ the remainder key fragment} is generated based on {TAIL ~ the k−1 key segments}; and 
is generated based on
storing a private key in k key fragments that include k−1 key segments and a remainder key fragment , where k is greater than 2 , wherein the k−1 key segments are generated as random key segments , and wherein { HEAD ~ the remainder key fragment } is generated based on { TAIL ~ the k−1 key segments } ; and
OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH OTH HEAD HEAD HEAD HEAD HEAD HEAD HEAD HEAD REL REL REL REL TAIL TAIL TAIL TAIL TAIL TAIL TAIL TAIL OTH OTH