群、环和域的关系 (Al)Closure under addition If a and b belong to s, then a+ b is also in (A2) Associativity of ad a+(6+c)=(a+b)+cfor all a, b, c in S (A3)Additive identity There is an element o in r such that a+0=0+a=a for all a in s (A4)Additive inverse For each a in s there is an element-a in s such that a+(a)=(a)+a=0 AcE a+b=b+a for all a b in s (MI)Closure under multiplication: If a and b belong to S, then ab is also in S (M2) Associativity of multiplication: a(bc)=(ab)c for all a, b, c in S (M3)Distributive laws a(b+c)=ab +ac for all a, b, cin s (a +b)c= ac bc for all a, b, c in s (M4) Commutativity of multiplication: ab= ba for all a, b in S (MS) Multiplicative identity There is an element 1 in S such that a1=la= a for all a in s (M6) No zero divisors If a. b in s and ab=0 then either a=or b=0 (MT) Multiplicative inverse If a belongs to s and a 0, there is an element al in S such that acl=ala=1 Figure 4.1 Group, Ring, and Field mfy@ustc.edu.cn 现代密码学理论与实践 9/55mfy@ustc.edu.cn 现代密码学理论与实践 9/55