Additive bit-serial algorithm for discrete logarithm modulo 2/sup k/ - Electronics Letters

Introduction and summary: Hardware capabilities for integer arithmetic generally include addition, multiplication, and division with precision k typically chosen as 16, 32 or 64. Multiplication and division are often implemented by recursive bit serial algorithms employing O(k) serial additions to avoid the size and power requirements of a large multiplier. The integer addition and multiplication operations realised are effectively ‘exact’ residue arithmetic operations with modulo 2. Hardware support for applications where fast residue arithmetic computation is desirable is typically limited to only residue addition and multiplication. There is a need to find efficiently implementable algorithms for other fundamental residue operations for the ‘hardware friendly’ modulus 2. Furthermore, for implementations where hardware support does not include a large multiplier, there is a particular need for additive bit-serial algorithms for these additional residue operations. The fundamental residue arithmetic operations supplementing residue addition and multiplication of particular interest for feasibility of hardware implementation are: multiplicative inverse, powering (or exponentiation), and discrete logarithm. Following [1] we herein employ jnj2k 1⁄4 j to denote the congruence relation n j (mod 2) with the residue j satisfying 0 j 2 1. The discrete logarithm modulo 2 with logarithmic base 3dlg( j)1⁄4 e of an odd residue j, 1 j 2 1, is the minimum exponent e, when it exists, such that j3ej2k 1⁄4 j. Similarly, e1⁄4 dlg(b,M)( j) represents the discrete logarithm modulo M with logarithmic base b of j: jbejM 1⁄4 j. From [2–4], dlg( j) exists whenever j jj82 {1, 3}, and also 0 dlg( j) 2 2 1. Furthermore, for any odd residue j with 1 j 2 1, there is a unique sign, exponent pair (s, e) with s2 {0, 1}, 0 e 2 2 1 such that jð 1Þ 3ej2k 1⁄4 j ð1Þ