The least primitive roots mod p

Nelson Carella

doi:10.1515/jmc-2024-0017

Article Open Access

The least primitive roots mod p

Nelson Carella

Published/Copyright: April 1, 2025

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From the journal Journal of Mathematical Cryptology Volume 19 Issue 1

Abstract

Let p > 1 be a large prime number, and let ε > 0 be a small number. The established unconditional upper bounds of the least primitive root u ≠ ± 1 , v 2 in the prime finite field F p have exponential magnitudes u ≪ p 1 ⁄ 4 + ε . This note contributes a new result to the literature. It proves that the upper bound of the least primitive root has polynomial magnitude u ≤ ( log p ) 1 + ε unconditionally.

Keywords: primitive root; least primitive root; finite field; cryptographic algorithm; complexity theory

MSC 2010: 11A07; 11N05; 11N32

1 Introduction

The established unconditional upper bounds of the least primitive root u ≠ ± 1 , v 2 in the prime finite field F p seem to have exponential magnitudes u ≪ p 1 ⁄ 4 + ε , see [1, Theorem 3], and the established conditional upper bounds seem to have polynomial magnitudes u ≪ ( log p ) 6 + ε (see [2, Theorem 1.3]). Moreover, there is a partial result of the form u < 475 ( log p ) 8 ⁄ 5 for infinitely many primes (see [3, Theorem 3]). Finally, there is a conjectured upper bound of the form u ≪ ( log p ) ( log log p ) 2 – the heuristic appears in [4, Section 4]. This note proposes a new result on the theory of primitive roots in finite fields, and it proves the conjectured upper bound for all large primes unconditionally. This result sharpens both the established unconditional results and the established conditional results.

Theorem 1.1

Let p ≥ p 0 be a large prime number, and let ε > 0 be a small real number. Then, there exists a primitive root u ≠ ± 1 , v 2 in the prime finite field F p such that

u ≪ ( log p ) 1 + ε .

In practice, this result holds for prime numbers p that are significantly smaller than the explicit lower bound p ≥ p 0 > 2 2145 ≈ 1 0 645 estimated in Section 5. In fact, the numerical data in [5, Table 1] show that the least primitive roots modulo p have very small magnitudes O ( ( log p ) ( log log p ) 2 ) for all primes, as conjectured, even the least prime primitive roots have the same upper bound.

Given the prime factorization of the totient p − 1 , an application of Theorem 1.1 leads to a deterministic primitive root search algorithm of polynomial time complexity O ( ( log p ) c ) , where c > 1 is constant. In contrast, the established deterministic primitive root search algorithm seems to have exponential time complexity O ( p 1 ⁄ 4 + ε ) (see [6, Theorem]). Similar primitive root search algorithms are studied by Grossman [7] and a survey of the literature and the most recent primitive root search algorithm appear by the study by Shparlinski [8]. Another application of Theorem 1.1 leads to a deterministic primality test algorithm of polynomial time complexity O ( ( log p ) 4 + ε ) , and the technical details appear in previous studies [9], [10], and [11, p. 1]. In contrast, the established deterministic primality test algorithms have higher time complexities: the AKS algorithm has a running time complexity of O ( ( log p ) 15 ⁄ 2 ) arithmetic operations (see [12, Theorem 5.3]) and the Gaussian period algorithm has a running time complexity of O ( ( log p ) 6 ) arithmetic operations (see [13, Theorem 1]. Furthermore, this result can be used to construct other related deterministic cryptographic algorithms of polynomial time complexities, such as the squareroot mod p and signature schemes as in the study by Brier et al. [14], etc.

This innovation is made possible by a new representation of the characteristic function for primitive roots, described in Lemma 3.2. It is much simpler than the standard characteristic function for primitive roots in finite fields, described in Lemma 3.1. In addition, it significantly simplifies some of the standard analytic methods utilized in the theory of primitive roots; consult the relevant literature such as [15], [16], [17], [5], et al. to compare the differences. The foundation of the main result is covered in Sections 3–8. Last but not least, the proof of Theorem 1.1 appears in Section 9.

2 Notation

The set N = { 0 , 1 , 2 , 3 , … } denotes the set of nonnegative integers, the set Z = { … , − 2 , − 1 , 0 , 1 , 2 , … } denotes the set of integers, and the set P = { 2 , 3 , 5 , … } denotes the set of prime numbers. The letters p , q , r ∈ P usually denote arbitrary prime numbers, and the letters a , b , c , k , m , n ∈ N usually denote arbitrary integers.

The symbol log x = ln x denotes the natural logarithm.
Let f , g : [ x 0 , ∞ ] ⟶ R be a pair of functions and assume g ( x ) > 0 . The big O notation is defined by
(1) f ( x ) = O ( g ( x ) ) ⇔ ∣ f ( x ) ∣ ≤ c g ( x ) ,
for some constant c > 0 as x → ∞ .
The symbol ≪ is defined by
(2) f ( x ) ≪ g ( x ) ⇔ ∣ f ( x ) ∣ ≤ c g ( x ) ,
for some constant c > 0 as x → ∞ .

3 Representations of the characteristic function

The multiplicative order of an element in a finite field is defined by ord p u = min { k : u k ≡ 1 mod p } . An element u ≠ ± 1 , v 2 is called a primitive root if ord p u = p − 1 . The characteristic function Ψ : G ⟶ { 0 , 1 } of primitive elements is one of the standard analytic tools employed to investigate the various properties of primitive roots in cyclic groups G . Many equivalent representations of the characteristic function Ψ of primitive elements are possible. Two of these representations are investigated here.

3.1 Divisor-dependent characteristic function

The divisor-dependent characteristic function was developed about a century ago (see [18, Theorem 496], [16, Lemma 2.3], [19, p. 258]), et al. This characteristic function detects the multiplicative order of an element by means of the divisors of the totient p − 1 . The precise description is stated in the following.

Lemma 3.1

Let p ≥ 2 be a prime, and let χ be a multiplicative character of multiplicative order ord χ = d , where d ∣ p − 1 . If u ∈ F p is a nonzero element, then

Ψ ( u ) = φ ( p − 1 ) p − 1 ∑ d ∣ p − 1 μ ( d ) φ ( d ) ∑ ord χ = d χ ( u ) = 1 , if ord p ( u ) = p − 1 , 0 , if ord p ( u ) ≠ p − 1 ,

where μ : N ⟶ { − 1 , 0 , 1 } is the Mobius function.

There are a few variant proofs of this result, and these proofs are widely available in the literature. Almost every result in the theory of primitive roots in finite fields is based on this characteristic function, but sometimes written in different forms. This techniques was developed by Landau [18], Vinogradov, [20], Erdos [21]. An extension of this characteristic function to the finite ring Z ⁄ n Z is presented in [22, Lemma 4]. The main obstacle in this technique is the prime decomposition of the totients p − 1 .

3.2 Divisor-free characteristic function

A new divisors-free representation of the characteristic function of primitive element is developed here. It is completely independent of the prime decomposition of the totients p − 1 . It detects the multiplicative order ord p ( u ) ≥ 1 of the element u ∈ F p by means of the solutions of the equation τ n − u = 0 in F p , where u and τ are constants, and n ∈ R = { n < p : gcd ( n , p − 1 ) = 1 } is a variable. This is the original source of this result, and it is the product of many years of research in number theory and finite fields.

Lemma 3.2

Let p ≥ 2 be a prime, and let τ be a primitive root mod p and let ψ ≠ 1 be a nonprincipal additive character of order ord ψ = p . If u ∈ F p is a nonzero element, then

Ψ ( u ) = ∑ gcd ( n , p − 1 ) = 1 1 p ∑ 0 ≤ s ≤ p − 1 ψ ( ( τ n − u ) s ) = 1 , if ord p ( u ) = p − 1 , 0 , if ord p ( u ) ≠ p − 1 .

Proof

Set the additive character ψ ( s ) = e i 2 π a s ⁄ p , where a ∈ F p × . As the index n ∈ R = { n < p : gcd ( n , p − 1 ) = 1 } ranges over the integers relatively prime to φ ( p ) = p − 1 , the element τ n ∈ F p × ranges over the primitive roots modulo p . Accordingly, the equation a = τ n − u = 0 has a unique solution n ∈ R if and only if the fixed element u ∈ F p is a primitive root. This implies that the inner sum in

(3) ∑ gcd ( n , p − 1 ) = 1 1 p ∑ 0 ≤ s < p e i 2 π ( τ n − u ) s p = 1 , if ord p ( u ) = p − 1 , 0 , if ord p ( u ) ≠ p − 1 .

collapses to ∑ 0 ≤ s < p e i 2 π a s ⁄ p = ∑ 0 ≤ s < p 1 = p . Otherwise, if the element u ∈ F p is not a primitive root, then the equation a = τ n − u = 0 has no solution n ∈ R , and the inner sum in (3) collapses to ∑ 0 ≤ s < p e i 2 π a s ⁄ p = 0 , this follows from the geometric series formula ∑ 0 ≤ n ≤ N − 1 w n = ( w N − 1 ) ⁄ ( w − 1 ) , where w = e i 2 π a ⁄ p ≠ 1 and N = p .□

4 Results on exponential sums

The exponential sums of interest in this analysis are presented in this section.

Lemma 4.1

Let p ≥ 2 be a large prime, and let x ≤ p . If τ is a primitive root modulo p and a ∈ [ 1 , p − 1 ] , then

(4) ∑ 1 ≤ n ≤ x gcd ( n , p − 1 ) = 1 e i 2 π a τ n ⁄ p ≪ p 1 ⁄ 2 + δ ,

where δ > 0 is a small real number.

Proof

The complete proof appears in [23, Lemma 4.1].□

Lemma 4.2

Let p ≥ 2 be a large prime, and let x ≤ p . If τ is a primitive root modulo p and a ∈ [ 1 , p − 1 ] , then

(5) ∑ 1 ≤ n ≤ x gcd ( n , p − 1 ) = 1 e i 2 π a τ n ⁄ p = ∑ 1 ≤ n ≤ x gcd ( n , p − 1 ) = 1 e i 2 π τ n ⁄ p + O ( p 1 ⁄ 2 + δ ) ,

where δ > 0 is a small real number and the implied constant is independent of a ≠ 0 .

Proof

The complete proof appears in [23, Lemma 4.3].□

5 Lower bound of the totient function

The totient function is defined by φ ( n ) = # { m : gcd ( m , n ) = 1 } = n ∏ r ∣ n ( 1 − 1 ⁄ r ) , where r ≥ 2 varies over the prime divisors of n (see [24, Theorem 2.4]). This subsection provides a detailed proof of the lower bound of this function.

Lemma 5.1

If p is a large prime, then

φ ( p − 1 ) p ≫ 1 log log p .

Proof

For any prime p , the ratio φ ( p − 1 ) ⁄ p can be rewritten as a product over the prime

(6) φ ( p − 1 ) p = p − 1 p ⋅ φ ( p − 1 ) p − 1 = p − 1 p ∏ r ∣ p − 1 1 − 1 r ,

where r ≥ 2 ranges over the prime divisor of p − 1 . This step follows from the identity φ ( n ) ⁄ n = ∏ r ∣ n ( 1 − 1 ⁄ r ) , where r ≥ 2 ranges over the prime divisors of n . Since the number p − 1 has fewer than 2 log p prime divisors (see [25, Theorem 2.10]), let x = 2 log p . Then, an application of the lower bound of the product given in [26, Theorem 6.12] yields

(7) φ ( p − 1 ) p ≥ p − 1 p ∏ r ≤ 2 log p 1 − 1 r > p − 1 p ⋅ e − γ log ( 2 log p ) 1 − 0.2 ( log ( 2 log p ) ) 2 ≫ 1 log log p > 0 ,

where γ > 0 is the Euler constant.□

An alternative result for the lower bound of the ratio φ ( n ) ⁄ n appears in [25, Theorem 2.9].

Remark 5.1

The explicit lower bound p ≥ p 0 > 2 2145 ≈ 1 0 645 is derived from the explicit parameter x = 2 log p 0 > 2,973 for the totient product given in [26, Theorem 6.12] and the extreme value of the prime divisor counting function ω ( p − 1 ) ≤ 2 log p . However, on average, the order of ω ( p − 1 ) is significantly smaller, i.e., x = ( log log p ) 3 ⁄ 2 (see [27, Theorem 3.1] for more details). Thus, on average, the explicit lower bound is expected to be significantly smaller than p 0 > 1 0 645 .

6 Fibers and multiplicities result

The multiplicities of certain values occurring in the estimate of the error term E ( z ) are computed in this section.

Lemma 6.1

Let p be an odd prime, let z = ( log p ) 1 + ε , and let τ ∈ F p be a primitive root in the finite field F p . Define the maps

(8) α ( n , u ) ≡ ( τ n − u ) mod p and β ( a , b ) ≡ a b mod p .

Then, the fibers α − 1 ( m ) and β − 1 ( m ) of an element 0 ≠ m ∈ F p have the cardinalities

(9) # α − 1 ( m ) ≤ z − 1 and # β − 1 ( m ) = z ,

respectively.

Proof

Let R = { n < p : gcd ( n , p − 1 ) = 1 } . Given a fixed u ∈ [ 2 , z ] , the map

(10) α : R × [ 2 , z ] ⟶ F p , defined by α ( n , u ) ≡ ( τ n − u ) mod p ,

is one-to-one. This follows from the fact that the map n ⟶ τ n mod p is a permutation of the nonezero elements of the finite field F p , and the restriction map n ⟶ ( τ n − u ) mod p is a shifted permutation, and it maps the subset

(11) R ⊂ F p to R − u ⊂ F p ,

see [19, Chapter 7] for extensive details on the theory of permutation functions of finite fields. Thus, as ( n , u ) ∈ R × [ 2 , z ] varies, a value m = α ( n , u ) ∈ F p is repeated at most z − 1 times. Moreover, the premises no primitive root u ≤ z = ( log p ) 1 + ε implies that m = α ( n , u ) ≠ 0 . This verifies that the cardinality of the fiber is

(12) # α − 1 ( m ) = # { ( n , u ) ∈ R × [ 2 , z ] : m ≡ ( τ n − u ) mod p } ≤ z − 1 .

Similarly, given a fixed a ∈ [ 1 , z ] , the map

(13) β : [ 1 , z ] × [ 1 , p − 1 ] ⟶ F p , defined by β ( a , b ) ≡ a b mod p ,

is one-to-one. Here, the map b ⟶ a b mod p permutes the nonzero elements of the finite field F p . Thus, as ( a , b ) ∈ [ 1 , z ] × [ 1 , p − 1 ] varies, each value m = β ( a , b ) ∈ F p × is repeated exactly z times. This verifies that the cardinality of the fiber is

(14)□ # β − 1 ( m ) = # { ( a , b ) ∈ [ 1 , z ] × [ 1 , p − 1 ] : m ≡ a b mod p } = z .

7 Evaluation of the main term

An asymptotic formula for the main term M ( z ) is computed in this section.

Lemma 7.1

Let ε > 0 be a small real number. If p ≥ 2 is a large prime and u ≤ z = ( log p ) 1 + ε , then

∑ 2 ≤ u ≤ z 1 p ∑ 1 ≤ n ≤ p − 1 gcd ( n , p − 1 ) = 1 1 = φ ( p − 1 ) p ⋅ ( log p ) 1 + ε + O ( 1 ) .

Proof

The number of relatively prime integers n < p coincides with the values of the totient function. Thus, a routine rearrangement gives

(15) ∑ 2 ≤ u ≤ z 1 p ∑ 1 ≤ n ≤ p − 1 gcd ( n , p − 1 ) = 1 1 = φ ( p − 1 ) p ∑ 2 ≤ u ≤ z 1 = φ ( p − 1 ) p ( ( log p ) 1 + ε − 1 ) = φ ( p − 1 ) p ⋅ ( log p ) 1 + ε + O ( 1 ) ,

where φ ( p − 1 ) ⁄ p < 1 .□

8 Estimate of the error term

A nontrivial upper bound of the error term is computed in this section. To achieve this objective, the error term is partitioned as E ( z ) = E 0 ( z ) + E 1 ( z ) . The upper bound of the first term E 0 ( z ) for n < p ⁄ z is derived using geometric summation/sine approximation techniques, and the upper bound of the second term E 1 ( z ) for p ⁄ z ≤ n ≤ p is derived using exponential sums techniques.

Lemma 8.1

Let ε > 0 be a small real number. Suppose p ≥ 2 is a large prime and u ≤ z = ( log p ) 1 + ε . If there is no primitive root u ≤ z = ( log p ) 1 + ε , then

∑ 2 ≤ u ≤ z 1 p ∑ 1 ≤ n ≤ p − 1 gcd ( n , p − 1 ) = 1 ∑ 1 ≤ s ≤ p − 1 ψ ( ( τ n − u ) s ) ≪ ( log p ) ( log log p ) ,

where ψ ( s ) = e i 2 π k s ⁄ p , with 0 < k < p , is an additive character.

Proof

The product of a point ( a , b ) ∈ [ 1 , z ] × [ 1 , p ⁄ z ) satisfies a b < p . This leads to the partition [ 1 , p ⁄ z ) ∪ [ p ⁄ z , p ) of the index n , which is suitable for the sine approximation a b ⁄ p ≪ sin ( π a b ⁄ p ) ≪ a b ⁄ p for ∣ a b ⁄ p ∣ < 1 on the first subinterval [ 1 , p ⁄ z ) (21). Thus, consider the partition of the triple finite sum

(16) E ( z ) = ∑ 2 ≤ u ≤ z 1 p ∑ 1 ≤ n ≤ p − 1 gcd ( n , p − 1 ) = 1 ∑ 1 ≤ s ≤ p − 1 e i 2 π ( τ n − u ) s p = ∑ 2 ≤ u ≤ z 1 p ∑ 1 ≤ n < p ⁄ z gcd ( n , p − 1 ) = 1 ∑ 1 ≤ s ≤ p − 1 e i 2 π ( τ n − u ) s p + ∑ 2 ≤ u ≤ z 1 p ∑ p ⁄ z ≤ n ≤ p − 1 gcd ( n , p − 1 ) = 1 ∑ 1 ≤ s ≤ p − 1 e i 2 π ( τ n − u ) s p = E 0 ( z ) + E 1 ( z ) .

The first suberror term E 0 ( z ) is estimated in Lemma 8.2, and the second suberror term E 1 ( z ) is estimated in Lemma 8.3. Summing these estimates yields

(17) E ( z ) = E 0 ( z ) + E 1 ( z ) ≪ ( log z ) ( log p ) + z p 1 ⁄ 2 − δ ≪ ( log z ) ( log p ) ,

where δ > 0 is a small real number. This completes the estimate of the error term.□

Lemma 8.2

Let p ≥ 2 be large primes. If τ is a primitive root modulo p and there is no primitive root u ≤ z = ( log p ) 1 + ε , then

(18) E 0 ( z ) = ∑ 2 ≤ u ≤ z 1 p ∑ 1 ≤ n < p ⁄ z gcd ( n , p − 1 ) = 1 ∑ 1 ≤ s ≤ p − 1 e i 2 π ( τ n − u ) s p = O ( ( log z ) ( log p ) ) .

Proof

To apply the geometric summation/sine approximation techniques, the subsum E 0 ( z ) is partitioned as follows:

(19) E 0 ( z ) = ∑ 2 ≤ u ≤ z 1 p ∑ 1 ≤ n < p ⁄ z gcd ( n , p − 1 ) = 1 ∑ 1 ≤ s ≤ p − 1 e i 2 π ( τ n − u ) s p = ∑ 2 ≤ u ≤ z 1 p ∑ 1 ≤ n < p ⁄ z gcd ( n , p − 1 ) = 1 ∑ 1 ≤ s ≤ p ⁄ 2 e i 2 π ( τ n − u ) s p + ∑ p ⁄ 2 < s ≤ p − 1 e i 2 π ( τ n − u ) s p = E 0,0 ( z ) + E 0,1 ( z ) .

Now, a geometric series summation of the inner finite sum in the first term yields

(20) E 0,0 ( z ) = ∑ 2 ≤ u ≤ z 1 p ∑ 1 ≤ n < p ⁄ z gcd ( n , p − 1 ) = 1 ∑ 1 ≤ s ≤ p ⁄ 2 e i 2 π ( τ n − u ) s p = 1 p ∑ 2 ≤ u ≤ z ∑ 1 ≤ n < p ⁄ z gcd ( n , p − 1 ) = 1 e i 2 π ( τ n − u p ) ( p 2 + 1 ) − e i 2 π ( τ n − u ) p 1 − e i 2 π ( τ n − u ) p ≤ 1 p ∑ 2 ≤ u ≤ z ∑ 1 ≤ n < p ⁄ z gcd ( n , p − 1 ) = 1 2 ∣ sin π ( τ n − u ) ⁄ p ∣ ,

where the last line in (20) follows from the hypothesis that u is not a primitive root. Specifically, 0 ≠ τ n − u ∈ F p for any n ≥ 1 such that gcd ( n , p − 1 ) = 1 and any u ≤ z = ( log p ) 1 + ε . Similar estimations using geometric series summation/sine approximation appear in [28, Chapter 23]. Utilizing Lemma 6.1, the first term has the upper bound

(21) E 0,0 ( z ) = 1 p ∑ 2 ≤ u ≤ z ∑ 1 ≤ n < p ⁄ z gcd ( n , p − 1 ) = 1 2 ∣ sin π ( τ n − u ) ⁄ p ∣ ≪ 2 p ∑ 1 ≤ a ≤ z ∑ 1 ≤ b < p ⁄ z 1 ∣ sin π a b ⁄ p ∣ ≪ 2 p ∑ 1 ≤ a ≤ z ∑ 1 ≤ b < p p π a b ≪ ∑ 1 ≤ a ≤ z 1 a ∑ 1 ≤ b < p 1 b ≪ ( log z ) ( log p ) ,

where a b < p and ∣ sin π a b ⁄ p ∣ ≠ 0 since p ∤ a b . Similarly, the second term has the upper bound

(22) E 0,1 ( z ) = ∑ 2 ≤ u ≤ z 1 p ∑ 1 ≤ n < p ⁄ z gcd ( n , p − 1 ) = 1 ∑ p ⁄ 2 < s ≤ p − 1 e i 2 π ( τ n − u ) s p = 1 p ∑ 2 ≤ u ≤ z ∑ 1 ≤ n < p ⁄ z gcd ( n , p − 1 ) = 1 e i 2 π ( τ n − u ) p − e i 2 π ( τ n − u p ) ( p 2 + 1 ) 1 − e i 2 π ( τ n − u ) p ≤ 1 p ∑ 2 ≤ u ≤ z ∑ 1 ≤ n < p ⁄ z gcd ( n , p − 1 ) = 1 2 ∣ sin π ( τ n − u ) ⁄ p ∣ ≪ ( log z ) ( log p ) .

This is computed in the way as done in (20) to (21), mutatis mutandis. Summing (21) and (22) yields

(23)□ E 0 ( z ) = E 0,0 ( z ) + E 0,1 ( z ) ≪ ( log z ) ( log p ) .

Lemma 8.3

Let p ≥ 2 be a large prime. If τ is a primitive root modulo p and there is no primitive root u ≤ z = ( log p ) 1 + ε , then

(24) E 1 ( z ) = ∑ 2 ≤ u ≤ z 1 p ∑ p ⁄ z ≤ n ≤ p − 1 gcd ( n , p − 1 ) = 1 ∑ 1 ≤ s ≤ p − 1 e i 2 π ( τ n − u ) s p = O z p 1 ⁄ 2 − δ ,

where δ > 0 is a small real number.

Proof

The hypothesis that there is no primitive root such that τ n − u ≠ 0 in the finite field F p for any u ≤ z and gcd ( n , p − 1 ) = 1 , implies that E 1 ( z ) has a nontrivial upper bound. To determine an upper bound, rearrange the triple finite sum E 1 ( z ) and apply Lemma 4.2 to the new inner sum on the second line in the following:

(25) E 1 ( z ) = ∑ 2 ≤ u ≤ z 1 p ∑ p ⁄ z ≤ n ≤ p − 1 gcd ( n , p − 1 ) = 1 ∑ 1 ≤ s ≤ p − 1 e i 2 π ( τ n − u ) s p = 1 p ∑ 2 ≤ u ≤ z ∑ 1 ≤ s ≤ p − 1 e − i 2 π u s p ∑ p ⁄ z ≤ n ≤ p − 1 gcd ( n , p − 1 ) = 1 e i 2 π s τ n p = 1 p ∑ 2 ≤ u ≤ z ∑ 1 ≤ s ≤ p − 1 e − i 2 π u s p ∑ p ⁄ z ≤ n ≤ p − 1 gcd ( n , p − 1 ) = 1 e i 2 π τ n p + O ( p 1 ⁄ 2 + δ ) ,

where δ > 0 is a small real number. The application of Lemma 4.2 to the inner exponential sum on the second line of (25) removes the dependence on the variable s ≠ 0 in exchange for a simpler exponential sum and an error term, which are both independent of the variable s . Now, use the exact evaluation ∑ 1 ≤ s ≤ p − 1 e − i 2 π u s p = − 1 , take absolute value, and apply the triangle inequality:

(26) ∣ E 1 ( z ) ∣ ≤ 1 p ∑ 2 ≤ u ≤ z ∑ 1 ≤ s ≤ p − 1 e − i 2 π u s p ∑ p ⁄ z ≤ n ≤ p − 1 gcd ( n , p − 1 ) = 1 e i 2 π τ n p + O ( p 1 ⁄ 2 + δ ) ≪ 1 p ∑ 2 ≤ u ≤ z ∣ − 1 ∣ ∑ p ⁄ z ≤ n ≤ p − 1 gcd ( n , p − 1 ) = 1 e i 2 π τ n p + ∣ p 1 ⁄ 2 + δ ∣ ≪ 1 p ∑ 2 ≤ u ≤ z ∣ − 1 ∣ ∣ p 1 ⁄ 2 + δ ∣ ≪ z p 1 ⁄ 2 − δ ,

where the exponential sum estimate

(27) ∑ p ⁄ z ≤ n ≤ p − 1 gcd ( n , p − 1 ) = 1 e i 2 π τ n p ≪ p 1 ⁄ 2 + δ

follows from Lemma 4.1.□

9 Least primitive roots in finite fields

The determination of an upper bound for the smallest primitive root in the finite field F p is based on a new characteristic function for primitive roots in finite field F p introduced in Section 3. The proof is broken up into several lemmas proved in Sections 3–8.

Define the counting function

(28) N p ( z ) = # { u ≤ z : ord p u = p − 1 } .

Proof of Theorem 1.1

Let p ≥ p 0 be a large prime number, and let z = ( log p ) 1 + ε , where ε > 0 is a small number. Suppose, by contradiction, that the least primitive root u 0 > z and consider the sum of the characteristic function over the short interval [ 2 , z ] , i.e.,

(29) N p ( z ) = ∑ 2 ≤ u ≤ z Ψ ( u ) = 0 .

Replacing the characteristic function, Lemma 3.2, into the nonexistence equation (29) and expanding it yield

(30) N p ( z ) = ∑ 2 ≤ u ≤ z Ψ ( u ) = ∑ 2 ≤ u ≤ z 1 p ∑ 1 ≤ n ≤ p − 1 gcd ( n , p − 1 ) = 1 ∑ 0 ≤ s ≤ p − 1 ψ ( ( τ n − u ) s ) = ∑ 2 ≤ u ≤ z 1 p ∑ 1 ≤ n ≤ p − 1 gcd ( n , p − 1 ) = 1 1 + ∑ 2 ≤ u ≤ z 1 p ∑ 1 ≤ n ≤ p − 1 gcd ( n , p − 1 ) = 1 ∑ 1 ≤ s ≤ p − 1 ψ ( ( τ n − u ) s ) = M ( z ) + E ( z ) .

The main term M ( z ) , which is determined by a double finite sum over the trivial additive character ψ ( s ) = 1 , is computed in Lemma 7.1, and the error term E ( z ) , which is determined by a triple finite sum over the nontrivial additive characters ψ ( s ) = e i 2 π k s ⁄ p ≠ 1 , is computed in Lemma 8.1.

Substituting these evaluation and estimate yields

(31) N p ( z ) = ∑ 2 ≤ u ≤ z Ψ ( u ) = M ( z ) + E ( z ) = φ ( p − 1 ) p ( ( log p ) 1 + ε + O ( 1 ) ) + O ( ( log p ) ( log log p ) ) .

Applying Lemma 5.1 to the totient ratio φ ( p − 1 ) ⁄ p show that the main term in (31) dominates the error term:

(32) N p ( z ) ≫ 1 log log p ⋅ ( log p ) 1 + ε + O ( ( log p ) ( log log p ) ) ≫ ( log p ) 1 + ε ( log log p ) 1 + O ( log log p ) 2 ( log p ) ε > 0 ,

as p → ∞ . Clearly, this contradicts the hypothesis (29) for all sufficiently large prime numbers p ≥ p 0 . Therefore, there exists a small primitive root u 0 ≤ z = ( log p ) 1 + ε , quod erat inveniendum.□

In synopsis, this result proves that the smallest primitive roots satisfy the conjectured upper bound c ( p ) ( log p ) ( log log p ) 2 , where c ( p ) > 0 is a constant. Furthermore, the mean value of the constant c ( p ) over the set of primes seems to be bounded by the limit supremum:

(33) c ¯ = lim x → ∞ 1 π ( x ) ∑ p ≤ x c ( p ) ≤ limsup p → ∞ g ˆ ( p ) ( log p ) ( log log p ) 2 = e γ = 1.781072 … .

The limit supremum for prime primitive roots g ˆ is discussed in [4, Section 4]. Hence, on average, the smallest primitive root satisfies the mean explicit inequality

(34) g ( p ) ≤ g ˆ < 2 ( log p ) ( log log p ) 2 ,

where e γ < 2 , and this follows from (33). The numerical data in the study by McGown et al. [5] shows that (34) is true for all primes up to p = 6525032504501281 . For example, for the prime p = 6525032504501281 , there is an abundance of primitive roots (both composite and prime) bounded by the mean explicit upper bound (34):

(35) 417,431,467,473,479,499,521,527,556,571,581,602,617,691,695,769,801,817,821,823,829,834 ≤ e γ ( log p ) ( log log p ) 2 = 838.1936 < 2 ( log p ) ( log log p ) 2 = 941.2234 .

The closest explicit result in this direction seems to be the claim that g ( p ) < p 5 ⁄ 8 for all prime p > 1 0 22 , which is proved in [17, Corollary 1] using the standard characteristic function given in Lemma 3.1.

For large prime p , the determination of the specific constant c ( p ) > 0 , which depends on p , and the mean value c ¯ of these constants seem to be difficult problems in analytic and algebraic number theory.

Acknowledgement

The author would also like to thank the anonymous referees for the excellent comments and suggestions to improve the quality of this manuscript.

Funding information: Author states no funding involved.
Author contributions: The author accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: There is no conflict of interest of any form.
Data availability statement: Data sharing is not applicable to this article.

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Received: 2024-04-10

Revised: 2024-10-11

Accepted: 2024-12-12

Published Online: 2025-04-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/jmc-2024-0017

Keywords for this article

primitive root; least primitive root; finite field; cryptographic algorithm; complexity theory

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