Application of Partial Discrete Logarithms for Discrete Logarithm Computation

Abstract

A novel approach to constructing an algorithm for computing discrete logarithms, which holds significant interest for advancing cryptographic methods and the applied use of multivalued logic, is proposed. The method is based on the algebraic delta function, which allows the computation of a discrete logarithm to be reduced to the decomposition of known periodic functions into Fourier–Galois series. The concept of the “partial discrete logarithm”, grounded in the existence of a relationship between Galois fields and their complementary finite algebraic rings, is introduced. It is demonstrated that the use of partial discrete logarithms significantly reduces the number of operations required to compute the discrete logarithm of a given element in a Galois field. Illustrative examples are provided to demonstrate the advantages of the proposed approach. Potential practical applications are discussed, particularly for enhancing methods for low-altitude diagnostics of agricultural objects, utilizing groups of unmanned aerial vehicles, and radio geolocation techniques.

1. Introduction

Finite algebraic fields, commonly referred to as Galois fields, are increasingly integral to various information technologies, notably in cryptography [1,2,3], digital signal processing [4,5,6], and related domains. A pivotal operation in constructing computational algorithms within such fields is discrete logarithm computation, which establishes a correspondence between non-zero elements of a Galois field $GF\left(p^n\right)$ and the exponent of a designated primitive element [7,8].

We recall that Galois fields are finite commutative bodies, that is, algebraic structures containing a finite number of elements for which the operations of addition, subtraction, multiplication, and division (for non-zero elements) are defined. The simplest example of a Galois field is $GF\left(p\right)$, which represents the ring of residue classes of the integers modulo and prime number $p$.

The problem under consideration has been extensively investigated across multiple dimensions, particularly in the context of cryptographic applications [9,10,11]. Furthermore, as noted in [12], the computation of discrete logarithms in groups constitutes a fundamental challenge in cryptography. It is equally evident that this problem is intricately linked to advancements in quantum computing [13,14,15]. For example, reference [16] explored quantum circuits designed to address certain learning tasks that remain intractable for classical models, including state-of-the-art deep neural networks.

Several approaches have been developed to address this problem, including ones applicable to finite fields of fixed characteristic [17]. For finite fields of small characteristic, heuristic quasi-polynomial algorithms have been proposed [18]. Notable methods for computing discrete logarithms include Joux’s algorithm [19], the baby-step giant-step algorithm [20,21], and the Pohlig–Hellman algorithm [22,23], among others. In the realm of quantum computing, Shor’s algorithm holds particular interest [24,25].

A significant limitation of existing methods for computing discrete logarithms is the substantial computational resources required, particularly when operating over fields with a large number of elements. This poses challenges for their implementation in systems such as DSP modules, microcontrollers with cryptographic workloads, and similar architectures. Additionally, a fundamental aspect of this problem is highlighted in [26], which notes that the first practical public-key cryptosystem, the Diffie–Hellman key exchange algorithm, relies on the assumption that discrete logarithms are computationally hard. This assumption, characterized as a hypothesis, underpins the presumed security of various other public-key schemes and, as emphasized in [26], remains a formidable challenge. At the time of publication [26], this hypothesis was the subject of extensive debate, a situation that has persisted in subsequent research [27].

Consequently, the pursuit of solutions to this problem remains highly relevant, particularly in the development of algorithms for computing discrete logarithms that optimize computational resources and address specific tasks.

The aim of this study is to develop an algorithm for computing discrete logarithms that significantly simplifies the computations compared to previously known algorithms (Joux’s algorithm [19], the baby-step giant-step algorithm [20,21], and the Pohlig–Hellman algorithm [22,23], among others). The algorithm, proposed here for the first time, is based on the use of the algebraic delta function introduced in [28]. The use of this approach reduces the computation of discrete logarithms to the decomposition of known periodic functions into Fourier–Galois series, which are increasingly employed in various information technology applications, encompassing both binary [29,30] and non-binary fields [31]. A further simplification of the computations is associated with the concept of the partial discrete logarithm, which is introduced in this work for the first time. This concept is grounded in the close relationship between Galois fields and their complementary finite algebraic rings, as demonstrated in [32].

The primary advantage of the proposed approach lies in its ability to significantly reduce the number of operations required to compute discrete logarithms. Moreover, the explicit computation of discrete logarithms makes it possible to reduce the operations of important multivalued logics to explicit algebraic expressions. The solution of this problem has become particularly relevant in light of recent developments in various types of neuromorphic materials [33,34,35,36]. The specific features of such materials, which include, in particular, analogs of “memory cells” capable of existing in several distinct stable states [36,37,38], necessitate the transition to computational algorithms based on non-binary logic [39].

2. Fundamental Algebraic Relations

As noted in the Introduction, the foundation of the proposed approach is the algebraic delta function [28]. It is defined as follows:

$${\delta }_i\left(x\right)=1-{\left(x-x_i\right)}^{p-1},$$

(1)

where $x_i$ is a fixed element of the field $GF\left(p\right)$.

This function exhibits the following property:

$$\delta \left(x-x_i\right)=\left\{\begin{array}{c}1,x=x_i\\ 0,x\ne x_i\end{array}\right.,$$

(2)

which holds because any non-zero element of the Galois field $GF\left(p\right)$ satisfies the following equation:

$$1-x^{p-1}=0,$$

(3)

The term “algebraic delta function” is chosen by analogy with the Dirac delta function: function (1) is non-zero only at a single element of the underlying set, in the same way that the Dirac delta function is non-zero only at one specific value of its variable.

The convenience of employing function (1) lies, in particular, in the following: Operations of multivalued logic are typically represented in tabular form in the current literature [40,41]. The use of function (1) makes it possible to transform the tabular representation into an explicit algebraic expression.

Consider a function of one variable, $f\left(x\right)$, whose arguments and values are elements of the field $GF\left(p\right)$. Suppose this function is initially specified in tabular form, that is, for each value $x_i$ the value $f\left(x_i\right)$ is given. We demonstrate that the expression

$$F\left(x\right)={\sum }_{i=0}^{i=p-1}f\left(x_i\right)\delta \left(x-x_i\right)$$

(4)

solves the stated problem. Namely, it provides the transition from the tabular form to an explicit algebraic expression.

When a specific element $x_{i_0}$ of the corresponding Galois field is substituted into expression (4), all summands appearing in the sum in the right part of Formula (4) turn to zero, except the summand for which $i=i_0$ is satisfied.

Consequently,

$$F\left(x_{i_0}\right)=f\left(x_{i_0}\right).$$

(5)

It is evident that, for Galois fields, the function (1), which is also applicable to cases involving multiple variables [28], serves as an algebraic normal form [42] (known as the Zhegalkin polynomial [43] in alternative terminology).

Expression (1) requires computing rather large powers. It can be substantially simplified in terms of Fourier–Galois series [44]. Specifically, in the theory of algebraic fields, it has been established that [45]

$${\left(y\pm x\right)}^q=y^q\pm x^q$$

(6)

where $q$ is the characteristic of the field.

For fields $GF\left(p\right)$, the number $p$ coincides with the characteristic. Direct verification proves the validity of the equality:

$$y^p-x^p=\left(y-x\right)\left(y^{p-1}+y^{p-2}x+\dots +y{x}^{p-2}+x^{p-1}\right)$$

(7)

By substituting the ratio (6) in the right part of Formula (7), we get [44]

$${\left(y-x\right)}^{p-1}=y^{p-1}+y^{p-2}x+\dots +y{x}^{p-2}+x^{p-1}$$

(8)

The elements $x_i$ can be indexed arbitrarily. In particular, one may set $x_i={\theta }^i$, where $\theta$ is a primitive element of the field $GF\left(p\right)$. The powers of such an element [45] exhaust all non-zero elements of the field. Consequently, expression (4) can be reformulated as

$$F\left(x\right)=\left(1-x^{p-1}\right){\sum }_{k=0}^{k=p-1}f\left({\theta }^i\right)-{\sum }_{k=1}^{k=p-1}{x}^{p-1-k}{Q}_k$$

(9)

where

$$Q_k={\sum }_{l=1}^{l=p-1}f\left({\theta }^l\right){\theta }^{kl},$$

(10)

and sequences

$$w_m=\left(1,{\theta }^m,{\theta }^{2m},{\theta }^{3m},\dots ,{\theta }^{\left(p-2\right)m}\right)$$

(11)

form a complete orthogonal basis on the interval containing p − 1 cycles [46], i.e.,

$${\sum }_{j=0}^{j=p-2}{w}_{k_1}^{\left(j\right)}{w}_{k_2}^{\left(j\right)}=\left\{\begin{array}{c}1,k_1\equiv k_2\left(mod\left(p-1\right)\right)\\ 0,k_1\not\equiv k_2\left(mod\left(p-1\right)\right)\end{array}\right.$$

(12)

Therefore, the values $Q_k$ are Fourier–Galois transforms of functions $f\left({\theta }^l\right)$:

$$Q_k=\widehat{F}\left[f\left(x\right)\right](k)={\sum }_{l=1}^{l=p-1}f\left({\theta }^l\right){\theta }^{k·l}$$

(13)

The first term in (9) vanishes by virtue of (3). Therefore, the explicit computation of the function $f\left(x\right)$ for an arbitrary $x$ can be expressed through its Fourier–Galois transform. This transform can be computed when the values of $f\left(x\right)$ are initially provided in tabular form, as noted above.

Consequently, the computation of the function $F\left(x\right)$ reduces to evaluating the following inverse Fourier–Galois transform:

$$F\left(x\right)=-{\sum }_{k=1}^{k=p-1}{x}^{p-1-k}\widehat{F}\left[f\left(x\right)\right]\left(k\right)$$

(14)

It is noteworthy that (14) can also be derived directly from the forward and inverse Fourier–Galois transforms. However, its derivation via the algebraic delta function provides additional validation of its applicability for the intended purpose. In this work, (14) is employed to construct algorithms for computing discrete logarithms.

3. Results

3.1. General Formula for Computing Discrete Logarithms in the Field $GF\left(p\right)$

The operation of discrete logarithm computation is defined as the mapping:

$$x={\theta }^n\to n$$

(15)

where θ is a primitive element of the Galois field GF(p). Formula (15) includes a critical clarification: the mapping $x\to n$ is not unique, as the primitive element in a Galois field is not singular. Specifically, the number of such primitive elements is given by $\phi \left(n\right)$, where $\phi \left(n\right)$ denotes Euler’s totient function, representing the count of natural numbers less than or equal to n that are coprime with $n$.

For a specific primitive element, the discrete logarithm operation (15) is expressed by the following formula, derived from Equation (4):

$$Dl\left(x\right)={\sum }_{n=0}^{n=p-2}n\delta \left(x-{\theta }^n\right),$$

(16)

This formula constitutes the foundation of the algorithm developed for computing discrete logarithms. It already provides an explicit expression for the discrete logarithm. This fundamentally distinguishes the proposed approach from the analogs known in the literature (Joux’s algorithm [19], the baby-step giant-step algorithm [20,21], and the Pohlig–Hellman algorithm [22,23], among others), as it is valid for any Galois field without exception, thereby possessing the most general character.

However, the direct use of Formula (16) is not optimal. As follows from the subsequent discussion, it can be substantially simplified, including by employing either the expression for a geometric progression or Fourier–Galois transformations.

The method of simplifying computations based on Formula (16), using the formula for a geometric progression, is described below.

The summation in (16) extends to $n=p-2$. This reflects the next property: ${\theta }^{p-1}=1$; consequently, $Dl\left(x={\theta }^{p-1}\right)=0$. Formula (16) underscores that the discrete logarithm operation can be computed for arbitrary values of $x$. Correspondingly, Equation (13) transforms into

$$Q_k={\sum }_{n=1}^{p-2}n{\theta }^{nk}$$

(17)

We now demonstrate that the series (17) can be expressed as an explicit algebraic expression. First, we will consider the case where ${\theta }^{nk}\ne 1$. In this case, the series (17) can be rewritten as

$$Q_k={\sum }_{n=1}^{p-3}{\sum }_n^{p-2}{\theta }^{mk}+{\theta }^{\left(p-2\right)k},$$

(18)

The transition from Formula (17) to Formula (18) is based on the fact that multiplying a given quantity by an integer $n$ can be replaced by a sum in which this quantity appears $n$ times. The final term in Equation (18) is isolated because term ${\theta }^{\left(p-2\right)k}$ enters the corresponding sum with weight 1.

The derivations presented below make it possible to obtain an explicit expression for the discrete logarithm in the form of a series in powers of the element $x$. The analysis of the convergence of this series is not required, as the number of its terms is finite by definition.

Each term in Equation (18) can be reformulated to leverage a geometric progression as follows:

$${\sum }_n^{p-2}{\theta }^{mk}={\theta }^{nk}{\sum }_{m=0}^{p-2-n}{\theta }^{mk}$$

(19)

By applying the geometric progression formula to (19), we obtain

$${\sum }_n^{p-2}{\theta }^{mk}={\theta }^{nk}{\displaystyle \frac{1-{\theta }^{\left(p-1-n\right)k}}{1-{\theta }^k}}={\displaystyle \frac{{\theta }^{nk}-{\theta }^{\left(p-1\right)k}}{1-{\theta }^k}}={\displaystyle \frac{{\theta }^{nk}-1}{1-{\theta }^k}}$$

(20)

The last term in Equation (18) is also conveniently expressed in the form of (20):

$${\theta }^{\left(p-2\right)k}={\theta }^{\left(p-2\right)k}{\displaystyle \frac{1-{\theta }^k}{1-{\theta }^k}}={\displaystyle \frac{{\theta }^{\left(p-2\right)k}-1}{1-{\theta }^k}}$$

(21)

Thus, we derive

$$Q_k={\displaystyle \frac{1}{1-{\theta }^k}}\left[{\sum }_{n=1}^{p-2}\left({\theta }^{nk}-1\right)\right]$$

(22)

Summing $p-2$ units modulo 2 yields the field element −2:

$${\sum }_{n=1}^{p-3}1=p-2\equiv -2\left(modp\right)$$

(23)

Using the geometric progression formula again, we compute

$${\sum }_{n=1}^{p-2}{\theta }^{nk}={\theta }^k{\sum }_{n=0}^{p-3}{\theta }^{nk}={\theta }^k{\displaystyle \frac{1-{\theta }^{\left(p-2\right)k}}{1-{\theta }^k}}={\displaystyle \frac{{\theta }^k-1}{1-{\theta }^k}}=-1$$

(24)

Substituting (23) and (24) into (22), we obtain the following:

$$Q_k={\displaystyle \frac{1}{1-{\theta }^k}}$$

(25)

It remains to consider the case where ${\theta }^{nk}=1$, corresponding to the spectral component:

$$Q_{p-1}={\sum }_{n=1}^{p-2}n={\sum }_{n=1}^{p-1}n-\left(p-1\right)=-\left(p-1\right)$$

(26)

This equality, like those above, is defined modulo $p$. The right-hand side accounts for the fact that $p-1$ is even (except for the case $p=2$, which is of no interest). Consequently, the sum of all elements from 1 to $p-1$ can be partitioned into pairs, where the sum of each pair equals $p$. This sum is evidently zero modulo $p$. Furthermore:

$$\left(p-1\right)\equiv -1\left(modp\right)$$

(27)

Substituting (25) and (27) into (18), we derive the expression for the discrete logarithm as the following series:

$$Dl\left(x\right)=-1-{\sum }_{k=1}^{k=p-2}{\displaystyle \frac{x^k}{1-{\theta }^{\left(p-1-k\right)}}}$$

(28)

This series can be simplified to avoid computing the highest powers of the primitive element. Since $p-1$ is even, the series (28) includes a term containing ${\theta }^{{\displaystyle \frac{\left(p-1\right)}{2}}}$. As ${\left({\theta }^{{\displaystyle \frac{\left(p-1\right)}{2}}}\right)}^2=1$, this term is a square root of unity, which can only be +1 or −1. Thus, ${\theta }^{{\displaystyle \frac{\left(p-1\right)}{2}}}=-1$.

We obtain

$$Q_{{\displaystyle \frac{\left(p-1\right)}{2}}}=-{\displaystyle \frac{1}{1-{\theta }^{\frac{\left(p-1\right)}{2}}}}=-2^{-1}\equiv {\displaystyle \frac{\left(p-1\right)}{2}}(modp)$$

(29)

Because $p-2$ is odd, the remaining terms in (28) can be paired for $k>{\displaystyle \frac{\left(p-1\right)}{2}}$ using the following identity:

$${\displaystyle \frac{1}{1-{\theta }^{\left(p-1-k\right)}}}={\displaystyle \frac{{\theta }^k}{{\theta }^k-1}}$$

(30)

Applying (30) to the series (28) and incorporating (29), we derive the following:

$$Dl\left(x\right)={\displaystyle \frac{p-3}{2}}+{\sum }_{k=1}^{k={\displaystyle \frac{p-3}{2}}}{\displaystyle \frac{{\theta }^k{x}^k-x^{\left(p-1-k\right)}}{1-{\theta }^k}}$$

(31)

It is evident that the computation of the series (31) involves only powers of the primitive element up to ${\displaystyle \frac{p-3}{2}}$. However, computing higher powers according to (31) is far from optimal, although this formula may prove useful in applications related to the practical use of multivalued logic (this issue is discussed in the Discussion section). From a computational perspective, it is considerably more efficient to employ partial discrete logarithms. This concept is introduced in the present work for the first time, with its justification provided in the following section.

3.2. Concept of Partial Discrete Logarithms

The concept of a partial discrete logarithm, which allows for a substantial simplification of discrete logarithm computation both in analytic form and as a computational algorithm, can be defined as follows. The number $p-1$ is not prime and can thus be expressed as a product of powers of prime numbers:

$$p-1=p_1^{q_1}{p}_2^{q_2}\dots p_k^{q_k}$$

(32)

where $p_i$ are distinct primes and $q_i$ are their respective exponents. Consequently, any non-zero element of the Galois field $GF\left(p\right)$ can be represented as a product, as follows:

$$z=g_1^{s_1}{g}_2^{s_2}\dots g_k^{s_k}$$

(33)

where the exponents $s_i$ range from 0 to $p_i^{q_i}-1$, and the elements $g_i$ in the field satisfy

$${g_i}^{p_i^{q_i}}=1$$

(34)

Equation (34) indicates that the elements $g_i$ can be interpreted as roots of unity of degree $p_1^{q_1}$. The validity of (33) can be illustrated as follows [46]. Consider an element $z_0$ corresponding to the special case where $s_1=s_2=\dots =s_k=1$.

$$z_0=g_1{g}_2\dots g_k$$

(35)

The powers of this element, by virtue of (34), can be expressed as

$${\left(z_0\right)}^r=g_1^{r\left(mod{p}_1^{q_1}\right)}{g}_2^{r\left(mod{p}_2^{q_2}\right)}\dots g_k^{r\left(mod{p}_k^{q_k}\right)}$$

(36)

As $r$ varies from 1 to $p-1$, the exponents $r\left(mod{p}_s^{q_s}\right)$ cycle periodically with period $p_s^{q_s}$. Notably, the interval from 1 to $p-1$ contains an integral number of periods for each $s$. Because these periods are distinct, the $p-1$ distinct powers computed via (36) encompass all non-zero elements of the field. Thus, $z_0$ is a primitive element of the field.

An illustrative example for the field $GF\left(13\right)$ is provided in

Table 1. Illustration of the representation of elements in the field $GF\left(13\right)$ using Formula (38).

$\mathit{r}$	${\mathit{g}}_{\mathbf{1}}^{{\mathit{s}}_{\mathbf{1}}}$	${\mathit{g}}_{\mathbf{2}}^{{\mathit{s}}_{\mathbf{2}}}$	${\mathit{g}}_{\mathbf{1}}^{{\mathit{s}}_{\mathbf{1}}}{\mathit{g}}_{\mathbf{2}}^{{\mathit{s}}_{\mathbf{2}}}$
1	3	5	2
2	9	12	4
3	1	8	8
4	3	1	3
5	9	5	6
6	1	12	12
7	3	8	11
8	9	1	9
9	1	5	5
10	3	12	10
11	9	8	7
12	1	1	1

. For this field, Equation (32) takes the following form:

$$p-1=12=2^2·3$$

(37)

Correspondingly, Equation (33) becomes

$$z=g_1^{s_1}{g}_2^{s_2}$$

(38)

where, for this specific field, $g_1=3$, $g_2=5$, as $g_1^3=1;g_2^4=1$, which can be verified by direct computation with modulo 13. The exponents in (38) vary within the ranges $s_1=1,2,0;s_2=1,2,3,0$.

In Table 1, the first column lists the index $r$; the second and third columns list the powers of the elements $g_1$ and $g_2$, respectively; and the fourth column lists their product. It is evident that the set of elements obtained via (38) exhausts the set of all 12 non-zero elements of the field.

The quantities $s_i=r\left(mod{p}_i^{q_i}\right)$ can be interpreted as partial discrete logarithms. We demonstrate that their computation is significantly simpler than computing the full discrete logarithm.

The use of this term is determined by the following considerations. The discrete logarithm is defined in accordance with expression (15)—that is, it represents the exponent to which the primitive element of a Galois field must be raised in order to obtain a given field element. The primitive element, as expressed in Formula (35), can be decomposed into factors that correspond to roots of unity.

To obtain a specific element of the Galois field, each of these factors must be raised to a certain power, after which their product is computed. Thus, instead of a single number determined by Formula (15), several values are calculated. The inverse mapping $x\to s_i$ may be interpreted as the computation of a partial discrete logarithm. More precisely, a partial discrete logarithm is defined as the exponent to which an individual factor of the decomposition of a primitive element into components must be raised in order to yield the desired element of the Galois field.

Because for an element ${\theta }^r$ the partial discrete logarithm is a well-defined quantity, the same approach as in the derivation of Formula (16) can be employed for its computation. Specifically, the partial discrete logarithm $s_i$ of the element ${\theta }^r$ is given by $s_i=r\left(mod{p}_i^{q_i}\right)$.

Applying Equation (4), the expression for the partial discrete logarithm $s_i$ is derived as

$$S_i\left(x\right)={\sum }_{m=1}^{m=p-1}{s}_i\left({\theta }^m\right)\delta \left(x-{\theta }^m\right),i=1,2,\dots ,k$$

(39)

where the functions $s_i\left({\theta }^m\right)$ are known and, over the interval from 1 to $p_i^{q_i}$, are given by

$$s_i\left({\theta }^m\right)=\left\{\begin{array}{c}m,m\le p_i^{q_i}-1\\ 0,m=p_i^{q_i}-1\end{array}\right.,$$

(40)

The functions in Equation (40) exhibit periodic continuation. For the specific case of $GF\left(13\right)$, Equation (39) takes the form

$$S_{1,2}\left(x\right)={\sum }_{m=1}^{m=12}{s}_{1,2}\left({\theta }^k\right)\delta \left(x-{\theta }^k\right)$$

(41)

As follows from the preceding discussion, Equation (39) can be expressed through a Fourier–Galois series representation:

$$S_i\left(x\right)=-{\sum }_{n=1}^{n=p-1}{x}^{p-1-n}{Q}_{ni}$$

(42)

where

$$Q_{ni}={\sum }_{l=1}^{l=p-1}{s}_i\left({\theta }^l\right){\theta }^{nl}$$

(43)

A key characteristic of the functions under consideration is their periodicity. It is precisely the periodic nature of the functions $s_i\left({\theta }^m\right)$ that allows for a substantial simplification of discrete logarithm computations, both in analytic derivations and in computational algorithms. We will demonstrate this.

Let us examine the Fourier–Galois spectrum of an arbitrary periodic function $R\left(m\right)$, with period $M$. It is evident that $M$ must be a divisor of $p-1$.

$$\left(w_n,R\right)={\sum }_{m=1}^{m=p-1}{\theta }^{mn}R\left(m\right)$$

(44)

Consider the case where ${\theta }^{Mn}\ne 1$. By grouping the terms of the series, we obtain

$$\left(w_n,R\right)={\sum }_{m=1}^{m=M}{\theta }^{mn}R\left(m\right){\sum }_{m_1=0}^{m_1={\displaystyle \frac{p-1}{M}}-1}{\theta }^{Mn{m}_1}$$

(45)

The second sum in this expression can be evaluated as the geometric progression

$${\sum }_{m_1=0}^{m_1={\displaystyle \frac{p-1}{M}}-1}{\theta }^{Mn{m}_1}={\displaystyle \frac{1-{\theta }^{n(p-1)}}{1-{\theta }^{Mn}}}=0$$

(46)

because ${\theta }^{n(p-1)}=1$ for any $n$.

Equation (46) indicates that the Fourier–Galois spectrum of a periodic function with period $M$ contains only components satisfying

$${\theta }^{Mn}=1$$

(47)

Let us explore this case. The condition in Equation (47) holds when

$$n=n_1{\displaystyle \frac{p-1}{M}},n_1=1,2,\dots ,M-1$$

(48)

This implies that the initial terms of the sequences forming the orthogonal basis in Equation (11), which correspond to non-zero spectral components $Q_{ni}$, can be expressed as

$$w_1(n_1)={\theta }^{n_1{\displaystyle \frac{p-1}{M}}},n_1=1,2,\dots ,M-1$$

(49)

In other words, these elements are roots of unity of degree ${\displaystyle \frac{p-1}{M}}$. For the field $GF\left(13\right)$, such elements correspond to sequences with the following indices (

Table 2. Elements of the sequences forming the orthogonal basis in Equation (11) for the field $GF\left(13\right)$.

N	n/nw	1	2	3	4	5	6	7	8	9	10	11	12
1	0	1	1	1	1	1	1	1	1	1	1	1	1
2	1	2	4	8	3	6	12	11	9	5	10	7	1
3	2	4	3	12	9	10	1	4	3	12	9	10	1
4	3	8	12	5	1	8	12	5	1	8	12	5	1
5	4	3	9	1	3	9	1	3	9	1	3	9	1
6	5	6	10	8	9	2	12	7	3	5	4	11	1
7	6	12	1	12	1	12	1	12	1	12	1	12	1
8	7	11	4	5	3	7	12	2	9	8	10	6	1
9	8	9	3	1	9	3	1	9	3	1	9	3	1
10	9	5	12	8	1	5	12	8	1	5	12	8	1
11	10	10	9	12	3	4	1	10	9	12	3	4	1
12	11	7	10	5	9	11	12	6	3	8	4	2	1

): for the component of the product in Equation (38), $g_1$, these are 4 and 8; for the component $g_2$, these are 3, 6, and 9.

Consequently, when condition (47) is satisfied, Equation (44) takes the form

$$\left(w_{n_1},R\right)={\displaystyle \frac{p-1}{M}}{\sum }_{m=1}^{m=M}{\theta }^{m{n}_1{\displaystyle \frac{p-1}{M}}}R\left(m\right)$$

(50)

In the case of computing partial discrete logarithms, the function $R\left(m\right)$ is known, and Equation (43) transforms into

$$Q_{n_{1}i}={\displaystyle \frac{p-1}{M}}{\sum }_{m=1}^{m=M}m{\theta }^{m{n}_1{\displaystyle \frac{p-1}{M}}}$$

(51)

Following the same reasoning that leads to Equation (31), we obtain the following expression for the amplitudes of the spectral components, which enables the computation of partial discrete logarithms:

$$Q_{n_{1}i}={\displaystyle \frac{p-1}{M}}{\displaystyle \frac{1}{1-{\theta }^{m{n}_1\frac{p-1}{M}}}}$$

(52)

An exception occurs for the case $n_1=0$, which corresponds to a sequence consisting solely of ones (zero powers of the primitive element). In this case,

$$Q_{0i}={\displaystyle \frac{p-1}{M}}{\sum }_{n_1=1}^{n_1=M-1}{n}_1$$

(53)

Combining the derived formulas, we obtain

$$S\left(x,M\right)=-{\displaystyle \frac{p-1}{M}}{\sum }_{n_1=1}^{n_1=M-1}{n}_1-{\sum }_{n_1=1}^{n_1={\displaystyle \frac{p-3}{M}}}{\displaystyle \frac{x^{{Mn}_1}}{1-{\theta }^{\left(p-1-M{n}_1\right)}}}$$

(54)

Formula (54) represents an explicit algebraic expression for the partial discrete logarithm.

However, the direct application of Equation (52) and/or (54) for Galois fields $GF\left(p\right)$ is not always convenient, although the analytic expression for the partial discrete logarithm is of interest for reducing operations of multivalued logic to an explicit algebraic form. The computational complexity arises from the fact that computing the inverse of a given element in the Galois field $GF\left(p\right)$ requires a specific algorithm. Depending on the nature of the problem being addressed, it may also be practical to employ a direct algorithm based on the straightforward summation of series of the form (17).

Nevertheless, it is evident that the computation of partial discrete logarithms is a significantly simpler procedure compared to the direct computation of a discrete logarithm. This is because, in the computation of partial discrete logarithms, a substantial portion of the spectral components reduce to zero. Specifically, for the field $GF\left(13\right)$, the number of non-zero spectral components in the computation of partial discrete logarithms is the sum of 3 and 4, whereas for the direct computation of a discrete logarithm, the number of components is 12, resulting in a difference of 5. This difference increases significantly as the value of $p$ grows. For instance, in the field $GF\left(61\right)$, the number of non-zero components for partial discrete logarithms is the sum of 4, 3, and 5 (while direct computation requires determining 60 spectral components), yielding a difference of 48.

The implementation of the algorithm for the direct computation of partial discrete logarithms is also relatively straightforward. Its block diagram is presented in Figure 1. This algorithm also includes the computation of a discrete logarithm from a set of partial logarithms. This method is discussed in Section 3.3.

This block diagram outlines the execution of the following operations. The “PN” block facilitates the input of numbers corresponding to the decomposition in Equation (32). If a prime number appears in the decomposition (32) with a certain exponent, the exponent of that number is used. For example, in the case of the field $GF\left(13\right)$, where $p-1=12$, the algorithm employs the numbers 4 and 3.

The “B” block generates the sequences constituting the orthogonal basis as defined by Equation (11). The input for this block’s computations is the value of the primitive element “PE”. For the field $GF\left(13\right)$, the primitive element is 2.

For the case of $GF\left(13\right)$, the sequences in Equation (11) are presented in

Table 3. Elements of the sequences in Equation (11) for the case of the field $GF\left(13\right)$.

n\N	1	2	3	4	5	6	7	8	9	10	11	12
0	1	1	1	1	1	1	1	1	1	1	1	1
1	2	4	8	3	6	12	11	9	5	10	7	1
2	4	3	12	9	10	1	4	3	12	9	10	1
3	8	12	5	1	8	12	5	1	8	12	5	1
4	3	9	1	3	9	1	3	9	1	3	9	1
5	6	10	8	9	2	12	7	3	5	4	11	1
6	12	1	12	1	12	1	12	1	12	1	12	1
7	11	4	5	3	7	12	2	9	8	10	6	1
8	9	3	1	9	3	1	9	3	1	9	3	1
9	5	12	8	1	5	12	8	1	5	12	8	1
10	10	9	12	3	4	1	10	9	12	3	4	1
11	7	10	5	9	11	12	6	3	8	4	2	1

. The first column of this table lists the powers of the primitive element n, which mark the starting point of each sequence (the corresponding numbers are shown in the rows of Table 3). The indices of the sequence elements N are displayed in the first row.

It is evident that there exist sequences with periods that are divisors of 12, i.e., the number of non-zero elements in the considered field. When decomposing the functions used for computing partial discrete logarithms (

Table 4. Characteristics of the periodicity of functions used for computing partial discrete logarithms Plog_1,2 and the discrete logarithm DLog.

N	1	2	3	4	5	6	7	8	9	10	11	12
Plog1	1	2	3	0	1	2	3	0	1	2	3	0
Plog2	1	2	0	1	2	0	1	2	0	1	2	0
DLog	1	2	3	4	5	6	7	8	9	10	11	0

) into the Fourier–Galois spectrum, only components with the same periodicity acquire non-zero values (

Table 5. Spectral components FPlog_1,2 and FDLog of the sequences FPlog_1,2 and the sequence DLog.

n	FPlog1,2	FPlog1,2	FDLog
0	5	12	1
1	0	0	12
2	0	0	4
3	11	0	11
4	0	6	6
5	0	0	5
6	7	0	7
7	0	0	9
8	0	8	8
9	3	0	3
10	0	0	10
11	0	0	2

, which, for comparison, also presents the values of the spectral components of the sequence DLog, used to compute the discrete logarithm). The computation of the Fourier–Galois spectrum for sequences of the form Plog_i is performed by the SFC block according to the scheme in Figure 1.

Partial discrete logarithms are computed by the FGT block, which effectively performs the inverse Fourier–Galois transform using the value of the element X, whose discrete logarithm is to be determined. Subsequently, based on the obtained values, the discrete logarithm is calculated in accordance with the methodology described in Section 3.3.

3.3. Computation of Discrete Logarithms via Partial Discrete Logarithms

Partial discrete logarithms, as defined by Equation (36), are given by

$$s_i=r\left(mod{p}_i^{q_i}\right)$$

(55)

where the value $r$ corresponds to the discrete logarithm. Consequently, the problem reduces to determining the index $r$ from the known set of quantities in Equation (55). This task is closely related to computations in the Residue Number System (RNS) and can be addressed as follows.

The value $r$ can be expressed in terms of the values $s_i$ as

$$r\equiv e_1{s}_1+e_2{s}_2+\dots +e_N{s}_N, mod(p-1)$$

(56)

where $e_i$—idempotent, mutually annihilating elements, and $s_i=0,1,2,\dots ,s_{im}$,

$$s_{im}={\displaystyle \frac{p-1}{p_i^{q_i}}}-1$$

(57)

This formula is directly expressed for the set of exponents $s_i$ corresponding to the exponent $r$, i.e., the discrete logarithm of a specific element in the considered Galois field. The idempotent elements are constructed according to the rule used, for instance, in [46]:

$$e_i={\alpha }_i{\prod }_{i\ne j}^N{p}_j^{q_j}$$

(58)

where ${\alpha }_i$—is an integer. These numbers are chosen to satisfy the condition

$$e_i{e}_i=1$$

(59)

By construction, it follows that

$$e_i{p}_i^{q_i}\equiv 0\left(P\right)$$

(60)

because any product of the form in Equation (38) includes the factor $p-1=p_1^{q_1}{p}_2^{q_2}\dots p_k^{q_k}$.

For the field $GF\left(13\right)$, used in the illustrative example, the idempotent elements in the context of Equation (56) are 9 and 4, as they satisfy

$$9·9=9 mod 12; 4·4=4 mod 12$$

(61)

Accordingly, in this case,

$$r\equiv 4·s_1+9·s_2$$

(62)

where $s_1=0,1,2;s_2=0,1,2,3$.

Thus, the concept of partial discrete logarithms is the basis of the proposed method of significantly simplifying the discrete logarithm computation for a range of Galois fields, including those relevant to practical applications.

Indeed, Formula (16) represents an analytic expression for computing the discrete logarithm in the most general case. It is applicable to any Galois field. However, its direct application involves considerable computational challenges. The calculations, including analytic ones, can be substantially simplified through the use of forward and inverse Fourier–Galois transforms. This approach necessitates computing the Fourier–Galois spectra of pre-defined periodic functions that determine the discrete logarithm, as given in Formula (17). Transitioning to the use of partial discrete logarithms provides a further significant simplification, given that a substantial portion of the spectral components of the functions defining the partial discrete logarithms vanish.

4. Discussion

The preceding discussion focused on Galois fields of the simplest form, $GF\left(p\right)$, but the proposed algorithm can be generalized, including to arbitrary fields $GF\left(p^m\right)$, that is, to fields obtained via algebraic extensions of the base field [47,48]. This generalization is achieved by employing a natural classification of basis functions of the form (12) that is generated by a specific field (or, more precisely, its primitive element) based on their periodicity. Among such fields, those of the form $GF\left(2^k\right)$, particularly $GF\left(2^8\right)$, are of significant interest, as they correspond to a widely adopted standard involving signal discretization with $2^8=256$ levels. According to the general theory of Galois fields, elements of such fields can be represented as

$$u=u_{k-1}{\theta }^{k-1}+u_{k-2}{\theta }^{k-2}+\dots +u_0$$

(63)

where the coefficients $u_n$ belong to the base field $GF\left(2\right)$, and $\theta$ is an element that constitutes a solution of an irreducible equation over the base field. Such equations, in particular, have no solutions in terms of the base field. For illustrative purposes, this element can be considered by analogy with the imaginary unit, that is, the root of the equation $x^2+1=0$, which has no solution in real variables. In this context, the element $\theta$ can be interpreted as a logical imaginary unit [49].

The possibility of using representation (63), as established in [28], implies that any operations performed on digital signals generated by electronic circuits designed for standard binary logic can be reduced to operations in multivalued logic. For the aforementioned standard, this corresponds to 256-valued logic. It is emphasized that this applies to all operations, including speech recognition, prediction generation, and similar tasks.

As demonstrated in [32], by excluding the zero element, it is possible to reduce p-valued logic to $p-1$-valued logic. This exclusion can be implemented electronically by adjusting the amplitude range of the processed signal. Without significant modifications to such circuits, one can transition, for example, from discretization using 256 levels to one using 255 levels. The “loss” of one level is inherently non-critical, as, with appropriate adjustment of the input signal gain, this level simply remains unused.

From the perspective of information theory, this implies that all information about the original signal is carried by a signal resulting from discrete logarithm computation. The same applies to the set of partial discrete logarithms. Consequently, any operations related to the processing of digital signal spectra can be reduced to the analysis of spectra of functions representing the discrete logarithm of the original signal as a function of time.

For fields $GF\left(2^k\right)$, this approach offers clear advantages in many cases of practical interest. These advantages are most evident in the specific case of $GF\left(2^8\right)$. For this field, transitioning to discrete logarithm computation corresponds to operations in the residue class ring modulo $256-1=255$. This ring allows the discrete logarithm to be expressed via Equation (56), which, for this specific case, takes the form

$$r\equiv 51·s_1+85·s_2+120·s_3, mod\left(255\right)$$

(64)

where $s_1=0,1,2$; $s_2=0,1,2,3,4$; and $s_3=0,1,2,\dots ,16$.

In deriving Equation (64), it is noted that 255 is the product of prime numbers

$$255=17·5·3$$

(65)

These prime numbers correspond to the idempotent elements (under multiplication modulo 255) appearing in Equation (64): $51=17·3$, $85=17·5$, and $120=4·5·3·2$. Consequently, any operations on signals reduced to this format (255 levels, excluding one level from the 256-level standard) can be expressed as operations on functions taking values in the Galois fields $GF\left(17\right)$, $GF\left(5\right)$, and $GF\left(3\right)$. A prominent example is speech recognition systems and their conversion to text format, and similar applications. Such systems are often based on spectral analysis [50,51], and transitioning to spectra derived from the Fourier–Galois transform significantly simplifies the algorithms. As highlighted in the cited work, such spectra also take values in the original field. Moreover, it is evident that transitioning to partial discrete logarithms in this case allows the use of fields of the form $GF\left(p\right)$ rather than their algebraic extensions, which would require the use of logical imaginary units and representations of the form (63). (An exception [52] is made for fields $GF\left(p\right)$ corresponding to quasi-Mersenne numbers $p=2^k+1$, for which a procedure for computing the discrete logarithm is proposed in the cited work.)

More broadly, the proposed approach is applicable to any problems involving discrete quantities of a fixed range. Any real digital signal (i.e., one divided into discrete levels) always varies within a finite amplitude range. Consequently, such a signal can be modeled by a function taking values in finite algebraic structures (Galois fields, finite algebraic rings, etc.). Similar considerations apply to tasks such as surface monitoring that inherently cover finite areas. In problems involving a transition to discrete coordinates, the coordinates of any specific “pixel” can be mapped to elements of a Galois field (or other finite algebraic structures). In particular, discrete coordinates on a plane or in space can be assigned to elements of the Galois field $GF\left(p^2\right)$ or $GF\left(p^3\right)$, respectively [53]. Another such example involves the use of finite algebraic rings [54]. The adoption of Galois coordinates enables the reduction of functions of two or three coordinates to single-variable functions. Furthermore, discrete Galois coordinates, or their analogs, address challenges that are intractable in continuous coordinate systems, such as the introduction of an analog to “three-dimensional” complex numbers. In contrast, continuous frameworks resolve such issues only in four dimensions, as exemplified by Hamilton’s quaternions [55]. In practical applications, the transition to discrete coordinates [53] is driven by needs such as enhancing algorithms for controlling groups of unmanned vehicles, a topic extensively explored in works like [56,57,58], some of which leverage fuzzy logic reducible to multivalued logic [59,60]. Additionally, it supports the development of novel information security approaches focused on determining the coordinates of signal sources [53,61].

It is noteworthy that numerous problems naturally lend themselves to discrete coordinate descriptions. For instance, as demonstrated in [62], specific conditions justify transitioning to descriptions of electromagnetic wave propagation in terms of discrete secondary sources. This approach enables the reduction of any electromagnetic wave transformer to a set of point transformers [62]. Such reductions are particularly relevant for addressing challenges in radio holography, a field currently attracting sustained research interest [63,64,65]. When combined with low-altitude sensing techniques for agricultural objects using UAV swarms, which are also under active development [66,67,68], radio holography methods offer unexpected applications. Namely, UAV swarms are among the most promising tools for low-altitude diagnostics of agricultural objects, including in the radio frequency range, which has long been used for remote sensing [69,70,71].

The potential of using UAV swarms is tied to fundamental factors. Specifically, the Earth’s surface acts as an interface between media with respect to radio waves, allowing the use of Fresnel equations. These equations indicate that the reflection coefficient increases with the angle of incidence. To utilize such waves (and particularly to perform angular scanning), receivers and transmitters must be spatially separated, which necessitates the use of UAV swarms. It is worth noting that such swarms can facilitate the diagnostics of subsurface objects (e.g., plant root systems and soil layer conditions, including moisture content). The possibility of solving problems of this kind by means of radio holography has also been proven in practice [72,73,74].

An important potential application of discrete logarithms lies in multivalued logic. Specifically, as shown in [28], a one-to-one correspondence can be established between the set of non-zero elements of a Galois field and the set of values in multivalued logic when the number of logic values is a prime number or a power of a prime. In this case, any operations in multivalued logic can be reduced to explicit algebraic expressions. More precisely, all operations in multivalued logic can be represented as addition and multiplication in Galois fields (a particular case being binary logic, where it is demonstrated that all logical operations reduce to logical addition and multiplication).

However, there exist important examples of multivalued logics that do not satisfy the above criterion (for instance, six-valued or ten-valued logics). Interest in such logics is partly driven by developments in neuromorphic materials [39], which exhibit substantial diversity [33,34,35]. Moreover, it is now recognized that the potential for developing computing systems based on binary logic and the von Neumann architecture has largely been exhausted [75], prompting active research into alternatives. These alternatives involve not only neuromorphic materials but also DNA-based computing systems [76,77], quantum computing [78,79], and so forth.

Operations in the above-mentioned types of logics can also be reduced to explicit algebraic expressions, but this requires transitioning from a Galois field to its associated algebraic ring [32]. An illustrative example is the reduction of operations in ten-valued logic to algebraic expressions, which is achieved by transitioning from the Galois field $GF\left(11\right)$ to its associated finite algebraic ring containing 10 elements. This transition is precisely realized through the computation of discrete logarithms of the elements of the original field. It is these logarithms that form the ring corresponding, for example, to ten-valued logic.

5. Conclusions

In this study, we developed a novel framework for computing discrete logarithms in Galois fields by introducing the concept of partial discrete logarithms and employing the algebraic delta function in combination with Fourier–Galois series expansions. Our main contributions can be summarized as follows:

• General formulation: We derived an explicit analytic expression for the discrete logarithm valid for any Galois field, including those obtained via algebraic extensions.
• Partial discrete logarithms: We introduced and formalized the notion of partial discrete logarithms, which enable a substantial reduction in the number of operations required compared with classical algorithms such as Joux’s, Pohlig–Hellman, etc.
• Practical relevance: We showed that the proposed method can be directly applied in digital signal processing and multivalued logic, with potential extensions to applications such as UAV-based monitoring, radio holography, and information security systems.

The results confirm that the use of partial discrete logarithms significantly simplifies the computation of discrete logarithms across a broad range of Galois fields. Furthermore, the generality of the approach ensures its applicability to fields of arbitrary size and structure, making it suitable for both theoretical cryptographic research and practical engineering tasks.

Future research will focus on optimizing the implementation of the proposed algorithms in resource-constrained environments such as microcontrollers and DSP modules, as well as exploring their integration with quantum-resistant cryptographic protocols and neuromorphic architectures based on multivalued logic.