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| ≤ n we cannot have t = ± n and so either t ∈,n

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Unformatted text preview: | ≤ n ; we cannot have t = ± n , and so either t ∈ { ,...,n- 1 } , or t + n ∈ { ,...,n- 1 } . Another interesting problem is exponentiation modulo n : given a ∈ { ,...,n- 1 } and a non- negative integer e , compute y = a e rem n . Perhaps the most obvious way to do this is to it- eratively multiply by a modulo n , e times, requiring time O ( e L ( n ) 2 ). A much faster algorithm, the repeated-squaring algorithm , computes y = a e rem n using just O ( L ( e )) multiplications modulo n , thus taking time O ( L ( e ) L ( n ) 2 ). This method works as follows. Let e = ( b ‘- 1 ··· b ) 2 be the binary expansion of e (where b is the low-order bit). For 0 ≤ i ≤ ‘ , define e i = ( b ‘- 1 ··· b i ) 2 . Also define, for 0 ≤ i ≤ ‘ , y i = a e i rem n , so y ‘ = 1 and y = y . Then we have e i = 2 e i +1 + b i (0 ≤ i < ‘ ) , 17 and hence y i = y 2 i +1 · a b i rem n (0 ≤ i < ‘ ) . This idea yields the following algorithm: y ← 1 for i ← ‘- 1 down to 0 do y ← y 2 rem n if b i = 1 then y ← y · a rem n output y It is clear that when this algorithm terminates, y = a e rem n , and that the running-time estimate is as claimed above. We close this chapter by observing that the Chinese Remainder Theorem (Theorem 2.6) can be made computationally effective as well. Indeed, by just using the formulas in the proof of that theorem, we see that given integers n 1 ,...,n k , and a 1 ,...,a k , with n i > 1, gcd( n i ,n j ) = 1 for i 6 = j , and 0 ≤ a i < n i , we can compute x such that 0 ≤ x < n and x ≡ a i (mod n i ) in time O ( L ( n ) 2 ), where n = Q i n i . We leave the details of this as an easy exercise. 18 Chapter 4 Abelian Groups This chapter reviews the notion of an abelian group. 4.1 Definitions, Basic Properties, and Some Examples Definition 4.1 An abelian group is a set G together with a binary operation ? on G such that 1. for all a,b ∈ G , a ? b = b ? a (commutivity property), 2. for all a,b,c ∈ G , a ? ( b ? c ) = ( a ? b ) ? c (associativity property), 3. there exists e ∈ G (called the identity element ) such that for all a ∈ G , a ? e = a (identity property), 4. for all a ∈ G there exists a ∈ G such that a ? a = e (inverse property). Before looking at examples, let us state some very basic properties of abelian groups that follow directly from the definition. Theorem 4.2 Let G be an abelian group with operator ? . Then we have 1. the identity element is unique, i.e., there is only one element e ∈ G such that a ? e = a for all a ∈ G ; 2. inverses are unique, i.e., for all a ∈ G , there is only one element a ∈ G such that a ? a is the identity. Proof. Suppose e,e are identities. Then since e is an identity, by the identity property in the definition, we have e ? e = e . Similarly, since e is an identity, we have e ? e = e . By the commutivity property, we have e ? e = e ? e . Thus, e = e ? e = e, and so we see that there is only one identity....
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