Primitive root modulo n

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In modular arithmetic, a branch of number theory, a number g is a primitive root modulo n if every number coprime to n is congruent to a power of g modulo n. In other words, g is a generator of the multiplicative group of integers modulo n. That is, for every integer a coprime to n, there is an integer k such that gka (mod n). Such k is called the index or discrete logarithm of a to the base g modulo n.

Gauss defined primitive roots in Article 57 of the Disquisitiones Arithmeticae (1801), where he credited Euler with coining the term. In Article 56 he stated that Lambert and Euler knew of them, but he was the first to rigorously demonstrate that primitive roots exist. In fact, the Disquisitiones contains two proofs: the one in Article 54 is a nonconstructive existence proof, while the other in Article 55 is constructive.

Elementary example[edit]

The number 3 is a primitive root modulo 7[1] because


\begin{array}{rcrcrcrcrcr}
3^1 &=&   3 &=& 3^0 \times 3 &\equiv& 1 \times 3 &=&  3 &\equiv& 3 \pmod 7 \\
3^2 &=&   9 &=& 3^1 \times 3 &\equiv& 3 \times 3 &=&  9 &\equiv& 2 \pmod 7 \\
3^3 &=&  27 &=& 3^2 \times 3 &\equiv& 2 \times 3 &=&  6 &\equiv& 6 \pmod 7 \\
3^4 &=&  81 &=& 3^3 \times 3 &\equiv& 6 \times 3 &=& 18 &\equiv& 4 \pmod 7 \\
3^5 &=& 243 &=& 3^4 \times 3 &\equiv& 4 \times 3 &=& 12 &\equiv& 5 \pmod 7 \\
3^6 &=& 729 &=& 3^5 \times 3 &\equiv& 5 \times 3 &=& 15 &\equiv& 1 \pmod 7 \\
\end{array}

Here we see that the period of 3k modulo 7 is 6. The remainders in the period, which are 3, 2, 6, 4, 5, 1, form a rearrangement of all nonzero remainders modulo 7, implying that 3 is indeed a primitive root modulo 7. Curiously, permutations created in this way (and their circular shifts) have been shown to be Costas arrays.

Definition[edit]

If n is a positive integer, the integers between 1 and n − 1 which are coprime to n (or equivalently, the congruence classes coprime to n) form a group with multiplication modulo n as the operation; it is denoted by Zn× and is called the group of units modulo n or the group of primitive classes modulo n. As explained in the article multiplicative group of integers modulo n, this group is cyclic if and only if n is equal to 2, 4, pk, or 2pk where pk is a power of an odd prime number.[2][3] A generator of this cyclic group is called a primitive root modulo n, or a primitive element of Zn×.

The order of (i.e. the number of elements in) Zn× is given by Euler's totient function \varphi\left(n\right). Euler's theorem says that aφ(n) ≡ 1 (mod n) for every a coprime to n; the lowest power of a which is congruent to 1 modulo n is called the multiplicative order of a modulo n. In particular, for a to be a primitive root modulo n, φ(n) has to be the smallest power of a which is congruent to 1 modulo n.

Examples[edit]

For example, if n = 14 then the elements of Zn× are the congruence classes {1, 3, 5, 9, 11, 13}; there are φ(14) = 6 of them. Here is a table of their powers modulo 14:

 x     x, x2, x3, ... (mod 14) 
 1 :   1 
 3 :   3,  9, 13, 11,  5,  1  
 5 :   5, 11, 13,  9,  3,  1
 9 :   9, 11,  1
11 :  11,  9,  1
13 :  13,  1

The order of 1 is 1, the orders of 3 and 5 are 6, the orders of 9 and 11 are 3, and the order of 13 is 2. Thus, 3 and 5 are the primitive roots modulo 14.

For a second example let n = 15. The elements of Z15× are the congruence classes {1, 2, 4, 7, 8, 11, 13, 14}; there are φ(15) = 8 of them.

 x     x, x2, x3, ... (mod 15) 
 1 :   1 
 2 :   2,  4,  8, 1  
 4 :   4,  1
 7 :   7,  4, 13, 1
 8 :   8,  4,  2, 1
11 :  11,  1
13 :  13,  4,  7, 1
14 :  14,  1

Since there is no number whose order is 8, there are no primitive roots modulo 15.

Table of primitive roots[edit]

This is Gauss's table of the primitive roots from the Disquisitiones. Unlike most modern authors he did not always choose the smallest primitive root. Instead, he chose 10 if it is a primitive root; if it isn't, he chose whichever root gives 10 the smallest index, and, if there is more than one, chose the smallest of them. This is not only to make hand calculation easier, but is used in § VI where the periodic decimal expansions of rational numbers are investigated.

The rows of the table are labelled with the prime powers (excepting 2, 4, and 8) less than 100; the second column is a primitive root modulo that number. The columns are labelled with the primes less than 97. The entry in row p column q is the index of q modulo p for the given root.

For example, in row 11, 2 is given as the primitive root, and in column 5 the entry is 4. This means that 24 = 16 ≡ 5 (mod 11).

For the index of a composite number, add the indices of its prime factors.

For example, in row 11, the index of 6 is the sum of the indices for 2 and 3: 21 + 8 = 512 ≡ 6 (mod 11). The index of 25 is twice the index 5: 28 = 256 ≡ 25 (mod 11). (Of course, since 25 ≡ 3 (mod 11), the entry for 3 is 8).

The table is straightforward for the odd prime powers. But the powers of 2, (16, 32, and 64) do not have primitive roots; instead, the powers of 5 account for one-half of the odd numbers less than the power of 2, and their negatives modulo the power of 2 account for the other half. All powers of 5 are ≡ 5 or 1 (mod 8); the columns headed by numbers ≡ 3 or 7 (mod 8) contain the index of its negative.

For example, modulo 32 the index for 7 is 2, and 52 = 25 ≡ –7 (mod 32), but the entry for 17 is 4, and 54 = 625 ≡ 17 (mod 32).

Primitive roots and indices (other columns are the indices of integers under respective column headings)
n root 2 3 5 7 11   13 17 19 23 29   31 37 41 43 47   53 59 61 67 71   73 79 83 89
3 2 1
5 2 1 3
7 3 2 1 5
9 2 1 * 5 4
11 2 1 8 4 7
13 6 5 8 9 7 11
16 5 * 3 1 2 1 3
17 10 10 11 7 9 13 12
19 10 17 5 2 12 6 13 8
23 10 8 20 15 21 3 12 17 5
25 2 1 7 * 5 16 19 13 18 11
27 2 1 * 5 16 13 8 15 12 11
29 10 11 27 18 20 23 2 7 15 24
31 17 12 13 20 4 29 23 1 22 21 27
32 5 * 3 1 2 5 7 4 7 6 3 0
37 5 11 34 1 28 6 13 5 25 21 15 27
41 6 26 15 22 39 3 31 33 9 36 7 28 32
43 28 39 17 5 7 6 40 16 29 20 25 32 35 18
47 10 30 18 17 38 27 3 42 29 39 43 5 24 25 37
49 10 2 13 41 * 16 9 31 35 32 24 7 38 27 36 23
53 26 25 9 31 38 46 28 42 41 39 6 45 22 33 30 8
59 10 25 32 34 44 45 28 14 22 27 4 7 41 2 13 53 28
61 10 47 42 14 23 45 20 49 22 39 25 13 33 18 41 40 51 17
64 5 * 3 1 10 5 15 12 7 14 11 8 9 14 13 12 5 1 3
67 12 29 9 39 7 61 23 8 26 20 22 43 44 19 63 64 3 54 5
71 62 58 18 14 33 43 27 7 38 5 4 13 30 55 44 17 59 29 37 11
73 5 8 6 1 33 55 59 21 62 46 35 11 64 4 51 31 53 5 58 50 44
79 29 50 71 34 19 70 74 9 10 52 1 76 23 21 47 55 7 17 75 54 33 4
81 11 25 * 35 22 1 38 15 12 5 7 14 24 29 10 13 45 53 4 20 33 48 52
83 50 3 52 81 24 72 67 4 59 16 36 32 60 38 49 69 13 20 34 53 17 43 47
89 30 72 87 18 7 4 65 82 53 31 29 57 77 67 59 34 10 45 19 32 26 68 46 27
97 10 86 2 11 53 82 83 19 27 79 47 26 41 71 44 60 14 65 32 51 25 20 42 91 18
    2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89

The sequence of smallest primitive roots (which is not the same as the sequence of primitive roots in the above table) may be found at (sequence A046145 in OEIS).

Arithmetic facts[edit]

Gauss proved[4] that for any prime number p (with the sole exception of p = 3), the product of its primitive roots is congruent to 1 modulo p.

He also proved[5] that for any prime number p, the sum of its primitive roots is congruent to μ(p – 1) modulo p where μ is the Möbius function.

For example

p = 3, μ(2) = –1. The primitive root is 2.
p = 5, μ(4) = 0. The primitive roots are 2 and 3.
p = 7, μ(6) = 1. The primitive roots are 3 and 5.
p = 31, μ(30) = –1. The primitive roots are 3, 11, 12, 13, 17 ≡ –14, 21 ≡ –10, 22 ≡ –9, and 24 ≡ –7.
Their product 970377408 ≡ 1 (mod 31) and their sum 123 ≡ –1 (mod 31).
3×11 = 33 ≡ 2
12×13 = 156 ≡ 1
(–14)×(–10) = 140 ≡ 16
(–9)×(–7) = 63 ≡ 1, and 2×1×16×1 = 32 ≡ 1 (mod 31).

Finding primitive roots[edit]

No simple general formula to compute primitive roots modulo n is known[citation needed]. There are however methods to locate a primitive root that are faster than simply trying out all candidates. If the multiplicative order of a number m modulo n is equal to \varphi\left(n\right) (the order of Zn×), then it is a primitive root. In fact the converse is true: If m is a primitive root modulo n, then the multiplicative order of m is \varphi\left(n\right). We can use this to test for primitive roots.

First, compute \varphi\left(n\right). Then determine the different prime factors of \varphi\left(n\right), say p1, ..., pk. Now, for every element m of Zn*, compute

m^{\varphi(n)/p_i}\mod n \qquad\mbox{  for } i=1,\ldots,k

using a fast algorithm for modular exponentiation such as exponentiation by squaring. A number m for which these k results are all different from 1 is a primitive root.

The number of primitive roots modulo n, if there are any, is equal to[6]

\varphi\left(\varphi\left(n\right)\right)

since, in general, a cyclic group with r elements has \varphi\left(r\right) generators.

If g is a primitive root modulo p, then g is a primitive root modulo all powers pk unless g p – 1 ≡ 1 (mod p2); in that case, g + p is.[7]

If g is a primitive root modulo pk, then g or g + pk (whichever one is odd) is a primitive root modulo 2pk.

Finding primitive roots modulo p is also equivalent to finding the roots of the (p-1)th cyclotomic polynomial modulo p.

Order of magnitude of primitive roots[edit]

The least primitive root gp modulo p (in the range 1, 2, ..., p − 1) is generally small.

Upper bounds[edit]

Burgess (1962) proved[8] that for every ε > 0 there is a C such that g_p < Cp^{\frac{1}{4}+\epsilon}.

Grosswald (1981) proved[8] that if p > e^{e^{24}}, then g_p < p^{0.499}.

Shoup (1990, 1992) proved,[9] assuming the generalized Riemann hypothesis, that gp = O(log6 p).

Lower bounds[edit]

Fridlander (1949) and Salié (1950) proved[8] that there is a positive constant C such that for infinitely many primes gp > C log p.

It can be proved[8] in an elementary manner that for any positive integer M there are infinitely many primes such that M < gp < p − M.

Applications[edit]

Primitive root modulo n is often used in cryptography, including the Diffie–Hellman key exchange scheme.

See also[edit]

Notes[edit]

  1. ^ http://www.brynmawr.edu/math/people/stromquist/numbers/primitive.html
  2. ^ Weisstein, Eric W., "Modulo Multiplication Group", MathWorld.
  3. ^ Primitive root, Encyclopedia of Mathematics
  4. ^ Gauss, DA, art. 80
  5. ^ Gauss, DA, art. 81
  6. ^ (sequence A010554 in OEIS)
  7. ^ This and the next assertion are in Cohen, p.26
  8. ^ a b c d Ribenboim 1996, p. 24
  9. ^ Bach & Shallit 1996, p. 254

References[edit]

The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.

External links[edit]