An algorithm is presented that produces an optimal radix-2 representation of an input integer n using digits from the set Dl,u={a Î mathbbZ:l £ a £ u}{D_{ell,u}={ain{mathbb{Z}}:ell le ale u}}, where ℓ ≤ 0 and u ≥ 1. The algorithm works by scanning the digits of the binary representation of n from left-to-right (i.e., from most-significant to least-significant); further, the algorithm is of the online variety in that it needs to scan only a bounded number of input digits before giving an output digit (i.e., the algorithm produces output before scanning the entire input). The output representation is optimal in the sense that, of all radix-2 representations of n with digits from D ℓ,u , it has as few nonzero digits as possible (i.e., it has minimal weight). Such representations are useful in the efficient implementation of elliptic curve cryptography. The strategy the algorithm utilizes is to choose an integer of the form d 2 i , where d Î Dl,u{d in D_{ell,u}} , that is closest to n with respect to a particular distance function. It is possible to choose values of ℓ and u so that the set D ℓ,u is unbalanced in the sense that it contains more negative digits than positive digits, or more positive digits than negative digits. Our distance function takes the possible unbalanced nature of D ℓ,u into account.
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机译：提出了一种算法，该算法使用集合D l，u = {aÎmathbbZ：l£a£u} {D_ {ell， u} = {ain {mathbb {Z}}：ell le ale u}}，其中ℓ≤0和u≥1。该算法通过从左到右扫描n的二进制表示形式的数字来工作（即，从最高有效到最低有效）；此外，该算法具有在线多样性，因为它在给出输出数字之前只需要扫描一定数量的输入数字（即，该算法在扫描整个输入之前就产生输出）。输出表示形式在以下方面是最佳的：在n的所有基数2表示形式中具有D ℓ，u 的数字中，它具有尽可能少的非零数字（即，具有最小的权重）。这样的表示在椭圆曲线密码术的有效实施中是有用的。该算法利用的策略是选择形式为d 2 i 的整数，其中dÎD l，u {d in D_ {ell，u}}，即关于特定距离函数最接近n。可以选择ℓ和u的值，以使集合D ℓ，u 不平衡，因为它包含的负数比正数多，或者正数比负数多。我们的距离函数考虑了D ℓ，u 的可能不平衡性质。
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