Algorithms for Determining Residues Modulo in a Complex Numerical Domain

Abstract

An important aspect of improving modern computer systems and their components is an increasing the speed of arithmetic calculations, including due to the use of new mathematical models and methods based on nonpositional residue number systems. The increase in the volume of processed data in modern computer systems leads to the additional risks and threats of unintentional failures and denials of service. This is especially important when building fault-tolerant critical information systems in which failure or denial of service can lead to catastrophic consequences. The article discusses arithmetic operations in the ring of residue classes. These techniques make it possible to implement fast and fault-tolerant computing for modern computer systems and telecommunication networks. We propose an algorithm for calculating the residues of integer data in a complex numerical domain. The algorithm is based on the use of the first fundamental Gauss theorem, which establishes an isomorphism between complex and real residues. Examples of determining the residues of integer data in a complex numerical domain are presented, which clearly demonstrate the constructiveness of the proposed techniques.

Keywords

• Computer Systems
• System of Residual Classes
• Reliable Calculations
• Arithmetic Operations

Introduction and Literature Review

The increase in the volume of processed and transmitted data in modern computer systems leads to the additional risks and threats of non-intentional failures and denials of service [1–3]. In this sense, an important direction of research is to increase the speed of computing devices based on the use of new mathematical models and computation methods [4–6], including non-positional residue number systems.

It is known, that in present time performance increasing of the integer data handling computer system and components (CSC), which are functioning in the binary positioning notation (PN), is connected, first of all, with increasing of the elements working frequencies and using of the formal synthesis patterns and methods, temporary multi parallel systems and programs [7–10]. At the same time it is theoretically and practically shown, that non-positional notation in the system of residue classes (SRC) usage allows fundamentally performance increasing and other CSC technical features improvement [11–16]. Besides the above-mentioned material, based on the research results, the fact of the efficient SRC usage in a hyper complex numeric area is important.

The integer rational numbers generalization is integer complex (Gaussian) numbers (CN) [8, 10]. Integer Gaussian numbers form a ring: its sum, difference, and multiplication are also (as the numbers in SRC) integer Gaussian numbers [7, 9].

Based on the SRC features, a set of the patentable components of the integer data handling computer system in the complex area was developed [17–20]. Nowadays, there is increasing the interest of the non-positional notation in SRC between information and telecommunication systems developers, which are implementing processes of forming, transferring and handling signals – physical data carriers, cryptographical data transforming, video data compression, etcetera [21–24].

The aim of the article is a consideration of the algorithm of the residues definition of the integer data in the complex numeric area. In particular, an algorithm of the real residue h defining of the integer complex number A = a + bi by complex modulo m = p + qi is considered. 2

Determining Residues Modulo in a Complex Numerical Domain In SRC there is the possibility for complex numbers to be presented in the form of their real residues, which means establishing the isomorphism between complex and real numbers residues. It gives a possibility of replacing arithmetical operations for integer Gaussian numbers to the same operations for the real numbers system by real modules, which are equal to norms of chosen complex SRC bases. In this aspect, there is an important task of transforming the number’s residue in SRC from a complex number area to the area of a real number. The task of transforming the number’s residue in SRC from a complex number area to the area of a real number is being solved by the way of the first fundamental Gauss’s law usage. Above-mentioned material leads to the first fundamental Gauss’s law. The law establishes isomorphism between complex and real residues.

Law formulation. By the given complex modulo m = p + qi , norm N of which equals to N =p2 + q2 and for which p and q are relative primes, each integer CN A = a + bi by complex modulo m is being compared to one and only one real residue from the set of numbers 0, N−1 , which means, that A ≡ h(mod m) .

Proof. It is known from the number theory, that for two relatively primes p and q it is possible to find such two integers u and v , that condition u ⋅ p + v ⋅ q =1 ( 1 ) is being met.

Showing the correctness of the following equation: i = u ⋅ p − v ⋅ q + m ⋅ (v + ui) . ( 2 )

Indeed

i = u ⋅ q − v ⋅ p + (p+ q ⋅ i) ⋅ (v + u ⋅ i) = u ⋅ q − v ⋅ p + ( p ⋅ v + p ⋅ u ⋅ i + q ⋅ v ⋅ i + q ⋅ u ⋅ i2 ) = = u ⋅ q − v ⋅ p + ( p ⋅ v + p ⋅ u ⋅ i + q ⋅ v ⋅ i − q ⋅ u) = = u ⋅ q − q ⋅ u − v ⋅ p + p ⋅ v + p ⋅ u ⋅ i + q ⋅ v ⋅ i = (u ⋅ p + v ⋅ q) ⋅ i Basing on the expression ( 1 ) i = i . Thus, equation ( 2 ) is correct.

If CN A = a + bi , then basing on the expression ( 2 ) there is: a + bi = a + b ⋅[u ⋅ q − v ⋅ p + m ⋅ (v + ui)] = a + (u ⋅ q − v ⋅ p) ⋅ b + m ⋅ (v ⋅ b + a ⋅ bi). ( 3 ) Defining h as the smallest positive real residue of number a + (u ⋅ q − v ⋅ p) ⋅ b by modulo N means that

Expression (4) is represented as an equation

h ≡ [a + (u ⋅ q − v ⋅ p) ⋅ b] mod N .

a + (u ⋅ q − v ⋅ p) ⋅ b = h + s ⋅ N Representing the expression ( 5 ) in the following form

h + s ⋅ N = h + s( p + qi) ⋅ ( p − qi) = h + m ⋅ ( p ⋅ s − q ⋅ si).

Then, based on expression ( 3 ), the equation is being fulfilled: a + bi = h + m ⋅ ( p ⋅ s − q ⋅ si) + m ⋅ (v ⋅ b + u ⋅ bi) = h + m ⋅[ p ⋅ s + v ⋅ b + (u ⋅ b − q ⋅ s)i], or in the form of congruence relation

(a + bi) ≡ h(mod m) .

Thus, it is proved, that the smallest complex residue x + vi of CN a + bi is said to be congruent modulo m with one and only one from the real numbers 0,1, 2,, N−1 .

Using the method of indirect proof defines that this number is unique. Assume, that there are two congruent relations as follows

(a + bi) ≡ h1 (mod m); (a + bi) ≡ h2 (mod m).

Basing on the feature of congruent relations there is

h1 ≡ h2 (mod m) or ( 4 ) ( 5 ) ( 6 ) which means

(h1 − h2 ) ≡ 0(mod m) , (h1 − h2 ) = m ⋅ (e + f ⋅ i) .

Expression ( 7 ) leads to fulfilling of the following equation

(m = p + qi) , (h1 − h2 ) = ( p + qi) ⋅ (e + fi) .

Multiplying both parts of the equation by the value p − qi leads to (h1 − h2 ) ⋅ ( p − qi) = ( p + qi) ⋅ ( p − qi) ⋅ (e + fi) , (h1 − h2 ) ⋅ ( p − qi) = ( p2 + q2 ) ⋅ (e + fi) , (h1 − h2 ) ⋅ ( p − qi) = N ⋅ (e + fi) , (h1 − h2 ) ⋅ p − (h1 − h2 ) ⋅ qi =N ⋅ e + N ⋅ fi . The last expression is equivalent to the next two real equation (h1 − h2 ) ⋅ p = N ⋅ e,  (h1 − h2 ) ⋅ q =−N ⋅ f . (h1 − h2 ) ≡ N ⋅ (e ⋅ u − f ⋅ v) (h1 − h2 ) ≡ 0(mod N ) . ( 7 ) ( 8 ) ( 9 ) Because CNs are equal, their real and imaginary parts are equal too. Multiplying the first equation of expression ( 8 ) by the value u and the second one by the value v , and then summing the results up leads to the following equation

(h1 − h2 ) ⋅ (u ⋅ p + v ⋅ q) = N ⋅ (e ⋅ u − f ⋅ v) .

Paying attention to an expression ( 1 ) u ⋅ p + v ⋅ q =1 , it follows, that or Since there is a suggestion h1, h2 < N , congruent relation ( 9 ) is possible only in the case h1 = h2 . Therefore, the possibility of existing the two different numbers h1 and h2 smaller than N , which would be congruent to a + bi modulo m , is eliminated. There is only one such number h , which is defined by the expression ( 4 ) and is represented in the form of congruent relation ( 10 ) [a + (u ⋅ q − v ⋅ p) ⋅ b] ≡ h(mod N ) . ( 10 ) In this case, there is usage of the following expression Z = (a + b ⋅ ρ ) , in which expression ρ = u ⋅ q − v ⋅ p , by using which the relation between complex and real residue by modulo m = p + qi is being established, is called as coefficient of isomorphism (CI). Thus, expression ( 10 ) is going to be presented in the following form Z ≡ h(mod N ) . ( 11 ) Data from the expressions ( 10 ) and ( 11 ) allows to define values of real residues Zi ≡ hi (mod N ) , (i =0, N−1) ,

p = u ⋅ q − v ⋅ p = u ⋅ 2 − v ⋅1 . corresponding to the smallest complex residues x + yi by modulo m = 1+ 2i . At first, there is defining the value of CI Values of v and u are defined by well-known in the number theory equation u ⋅ p − v ⋅ q =1 , meaning u ⋅1 − v ⋅ 2 =1 . By the way of selection u = −1 , q = 1 are being defined.

Thus, or For A =−1+ i .

For A = i .

For A =−1+ 2i .

For A = 2i .

p =(−1) ⋅ 2 −1⋅1 =−3 (−3) mod 5

=(N 2 = p2 + q2 =12 + 22 =5) .

Defining source values of the smallest real residues hi , isomorphic to the smallest complex residues, which are represented in table 2.

For A = 0 + 0i .

Z0 = a + bp = 0 + 0 ⋅ p = 0 . h0 = 0(mod 5) .

Z1 =−1+1⋅ (−3) =−4 . h1 = 1(mod 5) .

Z2 =0 +1⋅ (−3) =−3 . h2 = 2(mod 5) .

Z3 =−1+ 2 ⋅ (−3) =−1− 6 =−7 . h3 = 3(mod 5) .

Z4 =0 + 2 ⋅ (−3) =−6 . h4 = 4(mod 5) .

The results of the calculations of the smallest real remainders (residues) hi are in table 1.

Values of u and v are defined from the expression ( 1 ) by selecting values u, v . Thus, u = 1 and v = −2 . Checking the expression ( 1 ) shows, that

In this case, CI is equal to

1⋅ 5 + (−2) ⋅ 2 = 5 − 4 = 1 .

ρ = 1⋅ 2 − (−2) ⋅ 5 = 2 +10 = 12

Z = (16 + 7 ⋅ ρ ) = 16 + 7 ⋅12 = 100 .

Solving the congruent relation 100 ≡ h(mod 29) shows, that h ≡ 13(mod 29) . Thus, (16 + 7i) ≡ 13 mod(5 + 2i) .

Example 2. There is a congruent relation (1 + i) ≡ h mod(1 + 2i) to be solved. Or it is necessary to find the smallest real residue h of complex number (1 + i) by complex modulo (1 + 2i) .

In this case, GCD ( p, q) =( 1, 2 )

=1 , N = p2 + q2 =12 + 22 =5 .

A ≡ h(mod m) . h ≡ (a + b ⋅ ρ ) mod N .

Value of CI is equal to ρ = u ⋅ q − v ⋅ p = u ⋅ 2 − v ⋅1 , values of u and v are defined from the expression ( 1 )

u ⋅ p + v ⋅ q =1 , u ⋅1 + v ⋅ 2 =1 , i.e. u = −1 , v = 1 . and

Thus,

i.e.

ρ =(−1) ⋅ 2 −1⋅1 =−2 −1 =−3 , h = 1 + 1⋅ 2 = 3 , x + yi =4 + 2i, h =3 ,

(1 + i) ≡ 3 mod(1 + 2i) .

Examples 3 – 8 of defining the complex and real residues of integer complex number by complex modulo m = (1 + 2i) are going to be considered. Initial data for the examples solving is represented in table 2. m = (1 + 2i) , i.e. the aim is to find

Real residues h by

modulo N = p2 + q2 = 5

A ≡ (x + yi) mod m , ( a = 1 , b = 1 ; p = 1 , q = 2 ; N = 5 ).

Because of the famous equation, there is [8, 9] (1⋅1+1⋅ 2) ≡ (x ⋅1+ y ⋅ 2) mod 5,  (1⋅1−1⋅ 2) ≡ ( y ⋅1− x ⋅ 2) mod 5.

 3 = x + 2 y,  −1 =−2x + y. x = 3 − 2 y , −1 =−2 ⋅ (3 − 2 y) + y , −1 =−6 + 4 y + y , 5 y = 5 , y = 1 .

x = 3 − 2 y = 3 − 2 =1 ; x = 1 .

Answer: complex residue x + yi of CN A = 1+ i by complex modulo m = (1+ 2i) equals to complex number x + yi =1+ i .

Example 4. To define the smallest residue x + yi of CN A = 1+ i by complex modulo , i.e. the aim is to find a value

1+ i ≡ (x + yi) mod(1+ 2i) , ( a = 1 , b = 1 ; p = 1 , q = 2 ; N = 5 ).

Because of the famous equation, there is [7–10]

Γ = (1⋅1+1⋅ 2) mod 5 = 3 ; Γ′ = (1⋅1−1⋅ 2) mod 5 = (−1) mod 5 = 4 .

x + yi =3⋅1 − 4 ⋅ 2 + 4 ⋅1 + 3⋅ 2 i =− 5 + 10 i =−1 + 2i

5 5 5 5 .

Thus, the smallest residue x + yi of CN A = 1+ i by complex modulo m = (1+ 2i) equals to value x + yi =−1+ 2i . This solution can be represented in the form (1+ i) ≡ (−1+ 2i) mod(1+ 2i) .

Example 5. To solve congruent relation A ≡ h mod m of form

(1+ i) ≡ h mod(1+ 2i) , ( a = 1 , b = 1 ; p = 1 , q = 2 ; N = 5 ), expressions ( 1 ), ( 10 ), ( 11 ).

u ⋅ p + v ⋅ q =1 , u = −1 , u ⋅1+ v ⋅ 2 =1 . v = 1 . ρ = u ⋅ q − v ⋅ p .

Z = a + b ⋅ ρ , Z ≡ h mod N. ρ =(−1) ⋅ 2 −1⋅1 =−2 −1 =−3 .

Z =1+1⋅ (−3) =−2.

h ≡ (−2) mod 5 =3.

Thus, real residue h of CN A = 1+ i by complex modulo m = (1+ 2i) equals to value h = 3 .

Solution check. Achieved results should be checked. In example 4 there is the smallest complex residue (−1+ 2i) , and in example 5 there is real residue h = 3 . According to data from table 2 (−1+ 2i)  3 . Which is what it had to be shown.

Example 6. To define complex residue x + yi of CN A = 3 + 4i by complex modulo m = (1+ 2i) .

Using the famous equation [8, 9], there is system of congruent relations in form or Basing on the system of congruent relations there is system containing two linear equation because of (−2) =3 mod 5 .

x =11− 2 y , −2 ⋅ (11− 2 ⋅ y) + y =3 , −22 + 4 y + y =3 ,

5 y = 25 , y = 5 . x =11− 2 y =11−10 =1 .

Thus, complex residue x + yi of CN A = 3 + 4i by complex modulo m = (1+ 2i) equals to value

x + yi =1+ 5i .

Example 7. To define the smallest complex residue x + yi of CN A = 3 + 4i by complex modulo m = (1+ 2i) . N = 5 .

According to famous expressions, the smallest complex residue equals to value (x + yi) =Γ⋅ p − Γ′ ⋅ q + Γ′ ⋅ p + Γ ⋅ q i

N N Firstly, values Г and Г ′ need to be defined Γ = (a ⋅ p + b ⋅ q) mod N = (3⋅1+ 4 ⋅ 2) mod 5 = 11(mod 5) = 1 ; Γ′ = (b ⋅ p − a ⋅ q) mod N = (4 ⋅1 − 3 ⋅ 2) mod 5 = (−2) mod 5 = 3

In this case, there is

(x + yi) =

Thus, the smallest complex residue x + yi of CN A = 3 + 4i by complex modulo m = (1 + 2i) equals to value −1 + i .

Example 8. To define real residue h of CN A = 3 + 4i by complex modulo m = (1 + 2i) . N = 5 . The task can be formulated in another way. To solve a congruent relation (3 + 4i) ≡ h mod(1 + 2i) .

According to expression ( 11 ), there is (a + bρ ) ≡ h(mod N ) , where CI ρ is equal to ρ = u ⋅ q − v ⋅ p . Based on expression ( 1 ), values u and v are defined u ⋅ p + v ⋅ q =1 or u ⋅1 + v ⋅ 2 =1 .

So, the condition ( 1 ) will be fulfilled if u = −1 and v = 1 , i.e. (−1) ⋅1 + 1⋅ 2 =1 .

Basing on calculations

ρ =u ⋅ q − v ⋅ p =(−1) ⋅ 2 −1⋅1 =−3 . Z =(a + b ⋅ ρ ) =3 + 4 ⋅ (−3) =−9 .

There is (a + bρ ) ≡ h(mod N ) or (−9) ≡ 1(mod 5) , i.e. h = 1 .

Thus, there is the solution of a congruent relation (3 + 4i) ≡ 1mod(1 + 2i) .

Solution check. Achieved results should be checked. In example 7 there is the smallest complex residue (−1 + i) and in example 8 there is real residue h = 1 . According to data from table 2 (−1 + i)  1 . Which is what it had to be shown. 3

Conclusions

In the article algorithms of defining the residues by modulo in complex numeric area were considered. The main attention was paid to the algorithm of defining real residue of integer complex number by complex modulo, based on usage of first fundamental Gauss’s law. The examples of defining residues of integer data in the complex numeric area were provided. The results, achieved in the article, should be considered while implementing tasks and algorithms in SRC for the complex numeric area. Usage of represented methods contributes to performance increasing of SRC using for the quick implementation of integer operations in the complex numeric area. These computing techniques can be useful in various applications, for example, when processing data in complex computer systems, implementing reliable and fault-tolerant computers, and also for implementing cryptographic transformations [21, 25, 26].

The results can be used to build computer devices and components of fault-tolerant critical information systems. Increasing the speed of computing operations due to the use of non-positional residue number systems leads to a decrease in the risks of unintentional failures or denials of service of computer systems.

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