On the distribution of Low Hamming Weight products

Abstract

Jeffrey Hoffstein et al. (Discrete Appl. Math. 130:37–49, 2003) introduced the Low Hamming Weight products (LHWP) X=x_{1}x_{2}x_{3} as random exponent of elements in a group or a ring to improve the operational efficiency, where each x_{i} has Hamming Weight operatorname{Ham}(x_{i}) in its binary representation. The random power or multiple be used in many cryptographic constructions, such as Diffie–Hellman key exchange, elliptic curve ElGamal variants, and NTRU public-key cryptosystem. But their randomness is just a conjecture, which lacks of the security proof. The main purpose of this paper is using the analytic method and the properties of the character sums to prove the distribution of the Hamming weight products, which is related to their pseudorandomness and unpredictability. It is important to research the application of LHWP in cryptographic constructions. Our theory shows that the LHWP are exponentially close to the uniform distribution, namely, an attack on algorithm (Hoffstein et al. in Discrete Appl. Math. 130:37–49, 2003) needs polynomial time to reach exponentially close probabilities of success.