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Introduction to Number Theory 22 Index arithmetic Discrete

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					               Introduction to Number Theory
                     22. Index arithmetic
                Discrete logarithms
   Lemma 22.1. Suppose that m ∈ Z>0 has a primitive root r.
If a is a positive integer with (a, m) = 1, then there is a unique
integer x with 1 ≤ x ≤ φ(m) such that
                       r x ≡ a mod m.
Proof. By Th. 20.3., {r, r 2 , . . . , r φ(m) } is a reduced residue sys-
tem mod m. Therefore, if (a, m) = 1, then there is a unique
element in that set congruent to a mod m.
Definition 22.1 If m ∈ Z>0 has a primitive root r and a is a
positive integer with (a, m) = 1, then the unique integer x with
1 ≤ x ≤ φ(m) and r x ≡ a mod m is called the index (or discrete
logarithm) of a to the base r modulo m.
Notation. indr a.
Remark. In particular,
                    r indr a ≡ a mod m.
  Theorem 22.1. Let m be a positive integer with primitive root
r. If a, b are positive integers coprime to m and k is a positive in-
teger, then
(i) indr 1 ≡ 0 mod φ(m)
(ii) indr (ab) ≡ indr a + indr b mod φ(m)
(iii) indr ak ≡ k · indr a mod φ(m)
Proof. (i) Euler’s theorem implies that r φ(m) ≡ 1 mod m.
Therefore, indr 1 = φ(m) ≡ 0 mod φ(m).
(ii) By definition,
                          r indr a ≡ a mod m
                        r indr b ≡ b mod m and
                        r indr (ab) ≡ ab mod m.
Therefore,
      r indr (ab) ≡ ab ≡ r indr a r indr b = r indr a+indr b mod m.
Lemma 20.1 then implies that indr (ab) ≡ indr a+indr b mod φ(m).

(iii) Since, by (ii), indr (ak−1 a) ≡ indr ak−1 + indr a mod φ(m),
the result follows by induction on k.


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