Primitive $n$ -th roots of unity of finite fields

Theorem 6   For $n, p > 1$ , the finite field ${\mbox{${\mathbb{Z}}$}}/p{\mbox{${\mathbb{Z}}$}}$ has a primitive $n$ -th root of unity if and only if $n$ divides $p-1$ .

Proof. If ${\omega}$ is a a primitive $n$ -th root of unity in ${\mbox{${\mathbb{Z}}$}}/p{\mbox{${\mathbb{Z}}$}}$ then the set

$$\Omega \ = \ \{1, {\omega}, \ldots, {\omega}^{n-1} \}$$ (42)

forms a cyclic subgroup $H$ of the multiplicative group $G_{p-1}$ of ${\mbox{${\mathbb{Z}}$}}/p{\mbox{${\mathbb{Z}}$}}$ . By vertue of Lagrange's theorem (Theorem 5) the cardinality of $H$ divides that of $G_{p-1}$ . Since $G_{p-1}$ has $p-1$ elements, $n$ divides $p-1$ .

Conversly, it is known from finite field theory that $G_{p-1}$ is a cyclic group (even if $p$ is a power of a prime rather than a prime). Let ${\alpha}$ be a generator of this group, that is

$$G_{p-1} \ = \ \{ 1, {\alpha}, {\alpha}^2, \ldots, {\alpha}^{p-2} \}$$ (43)

Recall that ${\alpha}^{p-1} = 1$ from the little Fermat's theorem. Let $n > 1$ be an integer dividing $p-1$ and define

$${\omega} \ = \ {\alpha}^{(p-1)/n}.$$ (44)

Then we have

$${\omega}^n \ = \ 1.$$ (45)

For all $0 < k < n$ we have $k \frac{p-1}{n} < p-1$ so we have

$${\omega}^k = {\alpha}^{k(p-1)/n} \neq 1.$$ (46)

This shows that ${\omega}$ is a primitive $n$ -th root of unity. $\qedsymbol$

Example 6   Since $8 \ \mid (41 - 1)$ in ${\mbox{${\mathbb{Z}}$}}/41{\mbox{${\mathbb{Z}}$}}$ we have primitive $8$ -th root of unity. In fact the element $14 \in {\mbox{${\mathbb{Z}}$}}/41{\mbox{${\mathbb{Z}}$}}$ is such a root of unity.

Remark 5   Theorem 6 gives a necessary and sufficient condition for the existence of primitive $n$ -th roots of unity in ${\mbox{${\mathbb{Z}}$}}/p{\mbox{${\mathbb{Z}}$}}$ . But this does not give an algorithm to construct them.

We could use brute force. Given ${\mbox{${\mathbb{Z}}$}}/p{\mbox{${\mathbb{Z}}$}}$ , for every $n$ that we are interested in, for every $g \in G_{p-1}$ try if the following both statements hold:

• $g^n = 1$ ,
• for every prime divisor $t$ of $n$ we have $g^{n/t} \neq 1$ .

However we can reduce the complexity of this search by computing a generator of $G_{p-1}$ by means of Theorem 7 and then applying Relation (44).

Theorem 7   An element ${\alpha}$ of the multiplicative subgroup $G_{p-1}$ of the prime finite field ${\mbox{${\mathbb{Z}}$}}/p{\mbox{${\mathbb{Z}}$}}$ is a generator of $G_{p-1}$ iff for every prime factor $t$ of $p-1$ we have

$${\alpha}^{(p-1)/t} \ \neq \ 1$$ (47)

Proof. Assume that the condition holds and that ${\alpha}$ is not a generator of $G_{p-1}$ . Then there exists an integer $n$ such that

$$1 < n < p-1 \ \ \ \ {\rm and} \ \ \ \ {\alpha}^{n} = 1.$$ (48)

Let us choose $n$ minimum with this property. Then $H = \{ 1, {\alpha}, {\alpha}^2, \ldots, {\alpha}^{n-1} \}$ is a subgroup of $G$ . By virtue of Lagrange's theorem, the integer $n$ (the order of $H$ ) must divide ${p-1}$ . If $n$ is not a prime then there exists a prime $t$ dividing $n$ . Define

$${\beta} \ = \ {\alpha}^{n/t}.$$ (49)

Then $H' = \{ 1, {\beta}, {\beta}^2, \ldots, {\beta}^{t-1} \}$ is a subgroup of $G$ contradicting the choice of $n$ . Therefore $n$ must be a prime, which contradicts the assumption that the condition holds. $\qedsymbol$

Remark 6   For the purpose of multiplying polynomials with coefficients in ${\mbox{${\mathbb{Z}}$}}/p{\mbox{${\mathbb{Z}}$}}$ we are interested in primes $p$ and integers $n$ of the form $n = 2^k$ such that ${\mbox{${\mathbb{Z}}$}}/p{\mbox{${\mathbb{Z}}$}}$ possesses a primitive $n$ -th root of unity. By Theorem 6, the field ${\mbox{${\mathbb{Z}}$}}/p{\mbox{${\mathbb{Z}}$}}$ has such a root if there exists $q \in {\mbox{${\mathbb{Z}}$}}$ such that

$$p \ = \ q \, 2^k + 1$$ (50)

Such primes are called Fourier primes.

But not all numbers of the form $q \, 2^k + 1$ are prime (consider $q = k =2$ ). So how frequent are the primes among the numbers of the form $q \, 2^k + 1$ (for a given $k$ ). The answer is giving by the following theorem.

Theorem 8   Let $a$ and $b$ be two relatively prime integers. Then the number of primes

• in the arithmetic progression $a \, k + b$ for $k = 1, 2, \ldots$
• and not greater than a given integer $x$

is approximatively

$$\frac{x}{{\phi}(a) \, {\log}(x)}$$ (51)

where ${\phi}(a)$ is the Euler function at $a$ (the number of integers less than and relatively prime to $a$ ).

Remark 7   We apply the previous formula with $a = 2^k$ . All odd numbers less than $2^k$ are relatively prime to $2^k$ . (The even numbers are not relatively prime to $2^k$ .) Thus ${\phi}(2^k) = 2^{k-1}$ . Therefore there are approximatively

$$\frac{x}{2^{k-1} \, {\log}(x)}$$ (52)

Fourier primes less than a given integer $x$ .

Let $x = 2^{31}$ , which represents the usual size required for single precision integers. For $k = 20$ , there are approximatively 130 Fourier primes

• of the form $2^k \, q + 1$ ,
• and in the range $2^{20} \cdots 2^{31}$ .

This shows that there are 130 prime finite fields ${\mbox{${\mathbb{Z}}$}}/p{\mbox{${\mathbb{Z}}$}}$ such that

• $p$ fits in a machine integer (for a 32-bit processor)
• and allowing multiplication of polynomials in ${\mbox{${\mathbb{Z}}$}}/p{\mbox{${\mathbb{Z}}$}}[x]$ whose product is of degree less than $2^{20} = 1048576$ .

So, it looks like we have many possible primes for doing fast modular computations with polynomials. However primes $p$ such that $p^2$ fits in a machine integer are preferable in order to avoid the use of two machine words for multiplying two elements of ${\mbox{${\mathbb{Z}}$}}/p{\mbox{${\mathbb{Z}}$}}$ .

In the BasicMath library, one of the parents of libalgebra, nice primes are organized in tables. So there is a category for tables of primes!

macro SI == SingleInteger;

PrimesTableCategory(s: SI): Category ==  with {
        sizeBound: () -> SI;
        tableSize: () -> SI;
        rank: SI -> Partial(SI);
        maxPrime: () -> SI;
        minPrime: () -> SI;
        previousPrime: SI -> SI;
        nextPrime: SI -> SI;
        primes: () -> Generator(SI);
        FourierDegree: SI -> SI;
        getPrimeOfFourierDegree: SI -> SI;
        nextPrimeOfFourierDegree: (SI, SI) -> SI;
        unitsGenerator: SI -> SI;
        primitiveRootofUnity: (SI, SI) -> SI;
}

where

• PrimesTableCategory(s) specifies the operations that we expect from a table of primes $p$ less than $2^s$ . For obvious reasons, this category is restricted to primes that fit in a machine word.
• sizeBound() returns s.
• tableSize() returns the number of entries in the table.
• rank(p) returns tha rank of $p$ in the table if it lies in.
• FourierDegree(p) returns the biggest integer $r$ in the table such that $2^r$ divides $p-1$ .
• getPrimeOfFourierDegree(r) returns the smallest prime $p$ in the table such that FourierDegree(p) >= r, if any, otherwise 0 is returned.
• nextPrimeOfFourierDegree(n,r) returns the smallest prime $p$ in the table such that FourierDegree(p) = r and $p > n$ hold, otherwise 0 is returned.
• unitsGenerator(p) assuming that $p$ lies in the table returns a generator of the units group of the integers modulo $p$ .
• primitiveRootofUnity(p,n) assuming that $p$ lies in the table and assuming $0 < n \leq r$ where $r$ is FourierDegree(p), returns a primitive $2^n$ -root of unity of the units group of the integers modulo $p$ .

macro SI == SingleInteger;

FourierPrimesTableCategory(r: SI, s: SI): Category ==  with {
        FourierDegree: () -> SI
        sizeBound: () -> SI
        tableSize: () -> SI
        rank: SI -> Partial(SI)
        maxPrime: () -> SI
        minPrime: () -> SI
        previousPrime: SI -> SI
        nextPrime: SI -> SI
        primes: () -> Generator(SI)
        unitsGenerator: SI -> SI
        primitiveRootofUnity: (SI, SI) -> SI
}

whereFourierPrimesTableCategory(r,s) specifies the operations that we expect from the table of Fourier primes $p$ less than $2^s$ and such that there exists an odd integer $q$ satisfying $p-1=2^r q$ . Again we restrict to primes that fit in a machine word.

F6P15: FourierPrimesTableCategory(6,15) == add {
         local fd: SI == 6;
         local sb: SI == 15;
 -------------------------------------------------------------------- 
 -- Table (6,15)                                                  -- 
 -------------------------------------------------------------------- 

 -- Maximal size for a FFT: 2^6
 -- Number of 6-Fourier primes less than 2^15 is 58
	MAXINDEX: SingleInteger == 58;
 -- These Fourier primes are: 
	local primeList: Array SingleInteger := 
	   [193, 449, 577, 1217, 1601, 2113, 2753, 3137, 
	    4289, 4673, 4801, 5441, 5569, 5953, 6337, 6977, 
	    7489, 7873, 8513, 8641, 9281, 10177, 10433, 11329, 
	    11969, 12097, 13121, 13249, 13633, 14401, 14657, 15809, 
	    15937, 16193, 17729, 19009, 19777, 20161, 20929, 21313, 
	    21569, 22721, 23873, 24001, 25153, 25409, 25537, 25793, 
	    26177, 26561, 27073, 27329, 27457, 28097, 29633, 29761, 
	    30529, 32321];

 -- The corresponding units generators are: 
	local generatorList: Array SingleInteger := 
	   [5, 3, 5, 3, 3, 5, 3, 3, 
	    3, 3, 7, 3, 13, 7, 10, 3, 
	    7, 5, 5, 17, 3, 7, 3, 7, 
	    3, 5, 7, 7, 5, 11, 3, 3, 
	    7, 5, 3, 23, 11, 13, 7, 5, 
	    3, 3, 3, 14, 10, 3, 10, 3, 
	    3, 3, 5, 3, 7, 3, 3, 17, 
	    13, 6];

        FourierDegree(): SI == fd;
        sizeBound(): SI == sb;
        tableSize(): SI == MAXINDEX;
..............................................        
}

F7P15: FourierPrimesTableCategory(7,15) == add {
         local fd: SI == 7;
         local sb: SI == 15;
 -------------------------------------------------------------------- 
 -- Table (7,15)                                                  -- 
 -------------------------------------------------------------------- 

 -- Maximal size for a FFT: 2^7
 -- Number of 7-Fourier primes less than 2^15 is 29
	MAXINDEX: SingleInteger == 29;
 -- These Fourier primes are: 
	local primeList: Array SingleInteger := 
	   [641, 1153, 1409, 2689, 3457, 4481, 4993, 6529, 
	    7297, 9601, 9857, 10369, 11393, 12161, 13441, 13697, 
	    15233, 16001, 18049, 19073, 19841, 20353, 21121, 21377, 
	    26497, 28289, 29569, 30593, 31873];

 -- The corresponding units generators are: 
	local generatorList: Array SingleInteger := 
	   [3, 5, 3, 19, 7, 3, 5, 7, 
	    5, 13, 5, 13, 3, 3, 11, 3, 
	    3, 3, 13, 3, 3, 5, 19, 3, 
	    5, 6, 17, 3, 11];

        FourierDegree(): SI == fd;
        sizeBound(): SI == sb;
        tableSize(): SI == MAXINDEX;
...........................................................

F8P15: FourierPrimesTableCategory(8,15) == add {
         local fd: SI == 8;
         local sb: SI == 15;
 -------------------------------------------------------------------- 
 -- Table (8,15)                                                  -- 
 -------------------------------------------------------------------- 

 -- Maximal size for a FFT: 2^8
 -- Number of 8-Fourier primes less than 2^15 is 12
	MAXINDEX: SingleInteger == 12;
 -- These Fourier primes are: 
	local primeList: Array SingleInteger := 
	   [257, 769, 3329, 7937, 9473, 14081, 14593, 22273, 
	    23297, 26881, 30977, 31489];

 -- The corresponding units generators are: 
	local generatorList: Array SingleInteger := 
	   [3, 11, 3, 3, 3, 3, 5, 5, 
	    3, 11, 3, 7];

        FourierDegree(): SI == fd;
        sizeBound(): SI == sb;
        tableSize(): SI == MAXINDEX;
...........................................................

9P23: FourierPrimesTableCategory(9,23) == add {
         local fd: SI == 9;
         local sb: SI == 23;
 -------------------------------------------------------------------- 
 -- Table (9,23)                                                  -- 
 -------------------------------------------------------------------- 

 -- Maximal size for a FFT: 2^9
 -- Number of 9-Fourier primes less than 2^23 is 1092
        MAXINDEX: SingleInteger == 1092;
 -- These Fourier primes are: 
        local primeList: Array SingleInteger := 
           [7681, 10753, 11777, 17921, 23041, 26113, 32257,
...........................................................

F10P24: FourierPrimesTableCategory(10,24) == add {
         local fd: SI == 10;
         local sb: SI == 24;
 -------------------------------------------------------------------- 
 -- Table (10,24)                                                  -- 
 -------------------------------------------------------------------- 

 -- Maximal size for a FFT: 2^10
 -- Number of 10-Fourier primes less than 2^24 is 1087
        MAXINDEX: SingleInteger == 1087;
 -- These Fourier primes are: 
        local primeList: Array SingleInteger := 
           [13313, 15361, 19457, 25601,
...........................................................

F11P25: FourierPrimesTableCategory(11,25) == add {
         local fd: SI == 11;
         local sb: SI == 25;
 -------------------------------------------------------------------- 
 -- Table (11,25)                                                  -- 
 -------------------------------------------------------------------- 

 -- Maximal size for a FFT: 2^11
 -- Number of 11-Fourier primes less than 2^25 is 978
        MAXINDEX: SingleInteger == 978;
 -- These Fourier primes are: 
        local primeList: Array SingleInteger := 
           [18433, 
...........................................................

F12P26: FourierPrimesTableCategory(12,26) == add {
         local fd: SI == 12;
         local sb: SI == 26;
 -------------------------------------------------------------------- 
 -- Table (12,26)                                                  -- 
 -------------------------------------------------------------------- 

 -- Maximal size for a FFT: 2^12
 -- Number of 12-Fourier primes less than 2^26 is 972
        MAXINDEX: SingleInteger == 972;
 -- These Fourier primes are: 
        local primeList: Array SingleInteger := 
           [12289,
...........................................................