storing {TAIL ~ a private key} in k key fragments that include {HEAD ~ 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 the remainder key fragment is generated based on the k−1 key segments; and 
storing { TAIL ~ a private key } in k key fragments that include { HEAD ~ 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 the remainder key fragment is generated based on the k−1 key segments ; and
OTH TAIL TAIL TAIL TAIL TAIL TAIL TAIL OTH OTH OTH OTH OTH OTH HEAD HEAD HEAD HEAD HEAD HEAD HEAD 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 OTH