Degree of regularity for HFE Minus (HFE-)

MI: Global COE Program Education-and-Research Hub for Mathematics-for-IndustryグローバルCOEプログラム「マス･フォア･インダストリ教育研究拠点」In this paper, we prove a closed formula for the degree of regularity of the family of HFE- (HFE Minus) multivariate public key cryptosystems over a finite field of size q. The degree of regularity of the polynomial system derived from an HFE- system is less than or equal to ((q-1)(⌊log_q(D-1)⌋+a))/2+2 if q is even and r+a is odd, ((q-1)(⌊log_q(D-1)⌋+a+1))/2+2 otherwise. Here q is the base field size, D the degree of the HFE polynomial, r=⌊log_q(D-1)⌋+1 and a is the number of removed equations (Minus number). This allows us to present an estimate of the complexity of breaking the HFE Challenge 2: ・the complexity to break the HFE Challenge 2 directly using algebraic solvers is about 2^