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Any integer between 0 and p e 1 can be expressed as

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Unformatted text preview: Any integer between 0 and p e- 1 can be expressed as an e-digit number in base p ; for example, 49 y = (0 ··· 011) p . If we compute successive p-th powers of y modulo p e , then by Theorem 7.8 we have: y rem p e = (0 ··· 011) p y p rem p e = ( * ··· * 101) p y p 2 rem p e = ( * ··· * 1001) p . . . y p e- 2 rem p e = (10 ··· 01) p y p e- 1 rem p e = (0 ··· 01) p Here, “ * ” indicates an arbitrary digit. From this table of values, it is clear (c.f., Theorem 4.28) that [ y mod p e ] has order p e- 1 . That proves Theorem 7.4. Now consider Theorem 7.5. For e = 1 and e = 2, the theorem is clear. Suppose e ≥ 3. Consider the subgroup G ⊂ Z * 2 e generated by [5 mod 2 e ]. Expressing integers between 0 and 2 e- 1 as e-digit binary numbers, and applying Theorem 7.8, we have: 5 rem 2 e = (0 ··· 0101) 2 5 2 rem 2 e = ( * ··· * 1001) 2 . . . 5 2 e- 3 rem 2 e = (10 ··· 01) 2 5 2 e- 2 rem 2 e = (0 ··· 01) 2 So it is clear (c.f., Theorem 4.28) that [5 mod 2 e ] has order 2 e- 2 . We claim that [- 1 mod 2 e ] / ∈ G . If it were, then since it has order 2, and since any cyclic group of even order has precisely one element of order 2 (c.f., Theorem 4.24), it must be equal to [5 2 e- 3 mod 2 e ]; however, it is clear from the above calculation that 5 2 e- 3 6≡ - 1 (mod 2 e ). Let H ⊂ Z * 2 e be the subgroup generated by [- 1 mod 2 e ]. Then from the above, G ∩ H = { [1 mod 2 e ] } , and hence by Theorem 4.21, G × H is isomorphic to the subgroup G · H of Z * 2 e . But since the orders of G × H and Z * 2 e are equal, we must have G · H = Z * 2 e . That proves Theorem 7.5. 50 Chapter 8 Computing Generators and Discrete Logarithms in Z * p As we have seen in the previous chapter, for a prime p , Z * p is a cyclic group of order p- 1. This means that there exists a generator γ ∈ Z * p , such that for all α ∈ Z * p , α can be written uniquely as α = γ x for 0 ≤ x < p- 1; the integer x is called the discrete logarithm of α to the base γ , and is denoted log γ α . This chapter discusses some elementary considerations regarding the computational aspects of this situation; namely, how to efficiently find a generator γ , and given γ and α , how to compute log γ α . More generally, if γ generates a subgroup of Z * p of order q , where q | ( p- 1), and α ∈ h γ i , then log γ α is defined to be the unique integer x with 0 ≤ x < q and α = γ x . In some situations it is more convenient to view log γ α as an element of Z q . Also for x ∈ Z q , with x = [ a mod q ], one may write γ x to denote γ a . There can be no confusion, since if x = [ a mod q ], then γ a = γ a . However, in this chapter, we shall view log γ α as an integer....
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