Quadratic Residue Diffuser

While working through the design of an Auditorium, we took the opportunity to study the types of acoustic surfaces typical-

ly used to tune and temper the walls and ceilings of performance spaces. In addition to the overall geometry of the room,

the properties of these surfaces play an important role in how sound bounces around and within a space. In general, these

surfaces are categorized based on how sound interacts with them and fall under the three general headings -- reflective,

absorptive and diffusive. Reflective and absorptive elements fulfill definitive metrics, and are similarly specifically designed.

Reflective elements are generally designed as hard flat surfaces and are intended to bounce a sound with minimal affect to

its clarity and energy. Often made from softer materials, absorptive elements are very much the opposite and aim to capture

any sound that collides with it. If these two are understood as extremes of an interaction spectrum, diffusive elements land

somewhere within. Answering the question of where exactly a diffusive element lands presents interesting performative

design opportunities. Diffusion is intended to contribute to the enjoyment of a critical listening environment by disrupting

the clarity of a sound while maintaining a sound’s energy. Its effect is a sense of spaciousness. In earlier auditoriums and

theaters, diffusion occurred unsystematically through the use of ornament and decoration. As these spaces began to adopt

clean lines and remove ornament, other architectural elements needed to be developed to take up the diffusive function.

1^2 mod 256

149162536496481100121144169196225033681051441852281764113164217167313219306513220116891642416414522857144233681610971964114424910020964177361531613741290129413716153361776420910024914441196970161682331445722814564241164891620113265019313273162171641136417228185144105683302251961691441211008164493625169410149162536496481100121144169196225033681051441852281764113164217167313219306513220116891642416414522857144233681610971964114424910020964177361531613741290129413716153361776420910024914441196970161682331445722814564241164891620113265019313273162171641136417228185144105683302251961691441211008164493625169410

1^2 mod 64

14916253649017365716414330334411657361704936251694101491625364901736571641433033441165736170493625169410

1^2 mod 16

1490941014909410

1^2 mod 4

1010

1^2 mod 1

0 0 1

1^2 mod 2

0 1 4 1 0 1 4 1

1^2 mod 8

0 1 4 9 16 25 4 17 0 17 4 25 16 9 4 1 0 1 4 9 16 25 4 17 0 17 4 25 16 9 4 1

1^2 mod 32

014916253649648110012116416897033681051657100176411336891673465065473168936113641710057161056833097684116121100816449362516941014916253649648110012116416897033681051657100176411336891673465065473168936113641710057161056833097684116121100816449362516941

1^2 mod 128

Figure 2. Ray-tracing studies (enlarged image at end)

17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17

5^4 mod 32

0 29 8 15 0 9 24 27 0 21 8 7 0 1 24 19 0 13 8 31 0 25 24 11 0 5 8 23 0 17 24 3

5^3 mod 32

0 25 4 1 16 17 4 9 0 9 4 17 16 1 4 25 0 25 4 1 16 17 4 9 0 9 4 17 16 1 4 25

5^2 mod 32

5^1 mod 32

272217127229241914943126211611612823181383302520151050

4^4 mod 32

00000000000000000000000000000000

4^2 mod 32

160160160160160160160160160160160160160160160160

0 4 8 12 16 20 24 28 0 4 8 12 16 20 24 28 0 4 8 12 16 20 24 28 0 4 8 12 16 20 24 28

4^1 mod 32

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4^3 mod 32

2^6 mod 32

00000000000000000000000000000000

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2^5 mod 32

2^4 mod 32

160160160160160160160160160160160160160160160160

2^2 mod 32

4164041640416404164041640416404164041640

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

2^1 mod 32

0 8 0 24 0 8 0 24 0 8 0 24 0 8 0 24 0 8 0 24 0 8 0 24 0 8 0 24 0 8 0 24

2^3 mod 32

1^7 mod 32

3102101909070290270170150503025023013011010

1^5 mod 32

3101301102502305030170150290270907021019010

1^3 mod 32

31245038907241301181701524210198250232429027810

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

1^1 mod 32

0 1 4 9 16 25 4 17 0 17 4 25 16 9 4 1 0 1 4 9 16 25 4 17 0 17 4 25 16 9 4 1

1^2 mod 32

0 1 4 9 16 25 4 17 0 17 4 25 16 9 4 1 0 1 4 9 16 25 4 17 0 17 4 25 16 9 4 1

1^2 mod 32

0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1

1^4 mod 32

0 1 0 25 0 9 0 17 0 17 0 9 0 25 0 1 0 1 0 25 0 9 0 17 0 17 0 9 0 25 0 1

1^6 mod 32

0 15 30 13 28 11 26 9 24 7 22 5 20 3 18 1 16 31 14 29 12 27 10 25 8 23 6 21 4 19 2 17

15^1 mod 32

0 1 4 9 16 25 4 17 0 17 4 25 16 9 4 1 0 1 4 9 16 25 4 17 0 17 4 25 16 9 4 1

15^2 mod 32

0 15 24 21 0 19 8 25 0 23 24 29 0 27 8 1 0 31 24 5 0 3 8 9 0 7 24 13 0 11 8 17

15^3 mod 32

0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1

15^4 mod 32

0 13 26 7 20 1 14 27 8 21 2 15 28 9 22 3 16 29 10 23 4 17 30 11 24 5 18 31 12 25 6 19

13^1 mod 32

0 9 4 17 16 1 4 25 0 25 4 1 16 17 4 9 0 9 4 17 16 1 4 25 0 25 4 1 16 17 4 9

13^2 mod 32

3^4 mod 32

171610116170171610116170171610116170171610

0 21 8 23 0 1 24 3 0 13 8 15 0 25 24 27 0 5 8 7 0 17 24 19 0 29 8 31 0 9 24 11

13^3 mod 32

116170 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17

13^4 mod 32

0 19 24 1 0 7 8 21 0 27 24 9 0 15 8 29 0 3 24 17 0 23 8 5 0 11 24 25 0 31 8 13

11^3 mod 32

0 25 4 1 16 17 4 9 0 9 4 17 16 1 4 25 0 25 4 1 16 17 4 9 0 9 4 17 16 1 4 25

11^2 mod 323^2 mod 32

941716142502541161749094171614250254

0 11 22 1 12 23 2 13 24 3 14 25 4 15 26 5 16 27 6 17 28 7 18 29 8 19 30 9 20 31 10 21

11^1 mod 32

11617490

0 3 6 9 12 15 18 21 24 27 30 1 4 7 10 13 16 19 22 25 28 31 2 5 8 11

0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17

11^4 mod 32

14 17 20 23 26 29

3^1 mod 32

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8^4 mod 32

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8^3 mod 32

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8^2 mod 32

0 27 24 25 0 15 8 13 0 3 24 1 0 23 8 21 0 11 24 9 0 31 8 29

0 8 16 24 0 8 16 24 0 8 16 24 0 8 16 24 0 8 16 24 0 8 16 24 0 8 16 24 0 8 16 24

8^1 mod 32

0 19 24 17 0 7 8 5

3^3 mod 32

0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16

6^4 mod 32

0 24 0 8 0 24 0 8 0 24 0 8 0 24 0 8 0 24 0 8 0 24 0 8 0 24 0 8 0 24 0 8

6^3 mod 32

0 4 16 4 0 4 16 4 0 4 16 4 0 4 16 4 0 4 16 4 0 4 16 4 0 4 16 4 0 4 16 4

6^2 mod 32

0 19 0 9 0 15 0 5 0 11 0 1 0 7 0 29 0 3 0 25 0 31

0 6 12 18 24 30 4 10 16 22 28 2 8 14 20 26 0 6 12 18 24 30 4 10 16 22 28 2 8 14 20 26

6^1 mod 32

0 21 0 27 0 17 0 23 0 13

3^5 mod 32

7^1 mod 32

251811429221581261912530231692272013631241710328211470

7^2 mod 32

17425169410149162541701742516941014916254170

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7^3 mod 32

98190524150181102924702583021243101782701324230

0 0 0 0 0 0 0 0 0 0 0 0

16^4 mod 327^4 mod 32

116170171610116170171610116170171610116170171610

9^1 mod 32

231452819101241562920112251673021123261783122134271890

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9^2 mod 32

17425169410149162541701742516941014916254170

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

16^3 mod 32

0

9^3 mod 32

72429011810152450198902324130278170312421038250

9^4 mod 32

116170171610116170171610116170171610116170171610

0 0 0 0 0 0 0 0 0 0 0 0 0 0

10^1 mod 32

221222414426166281883020100221222414426166281883020100

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

16^2 mod 3210^2 mod 32

4164041640416404164041640416404164041640

10^3 mod 32

2408024080240802408024080240802408024080

0 16 0 16 0 16 0 16 0 16 0

10^4 mod 32

160160160160160160160160160160160160160160160160

16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16

16^1 mod 3212^1 mod 32

2082816424120208281642412020828164241202082816424120

12^2 mod 32

160160160160160160160160160160160160160160160160

12^3 mod 32

00000000000000000000000000000000

0 16 0 16 0 16 0 16 0

12^4 mod 32

00000000000000000000000000000000 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16

14^4 mod 32

14^1 mod 32

184228261230162206241028140184228261230162206241028140

14^2 mod 32

4164041640416404164041640416404164041640

14^3 mod 32

8024080240802408024080240802408024080240

1^2 mod 256

149162536496481100121144169196225033681051441852281764113164217167313219306513220116891642416414522857144233681610971964114424910020964177361531613741290129413716153361776420910024914441196970161682331445722814564241164891620113265019313273162171641136417228185144105683302251961691441211008164493625169410149162536496481100121144169196225033681051441852281764113164217167313219306513220116891642416414522857144233681610971964114424910020964177361531613741290129413716153361776420910024914441196970161682331445722814564241164891620113265019313273162171641136417228185144105683302251961691441211008164493625169410

1^2 mod 64

14916253649017365716414330334411657361704936251694101491625364901736571641433033441165736170493625169410

1^2 mod 16

1490941014909410

1^2 mod 4

1010

1^2 mod 1

0 0 1

1^2 mod 2

0 1 4 1 0 1 4 1

1^2 mod 8

0 1 4 9 16 25 4 17 0 17 4 25 16 9 4 1 0 1 4 9 16 25 4 17 0 17 4 25 16 9 4 1

1^2 mod 32

014916253649648110012116416897033681051657100176411336891673465065473168936113641710057161056833097684116121100816449362516941014916253649648110012116416897033681051657100176411336891673465065473168936113641710057161056833097684116121100816449362516941

1^2 mod 128

Figure 1. Array of profile variations and ray-tracing studies (enlarged image at end)

RESEARCH

After some research and discussions with consulting acousticians, we came upon the diffuser designs of physicist Manfred

Schroeder. Schroeder’s diffuser design is considered by some to be the most significant event in diffuser development.

While most diffusion came as the result of textures, ornament and general architectural irregularities -- all of which made

predictions about success difficult to be sure -- Schroeder used number sequences to dictate the relative dimensions of

articulations and verified how sound interacts. His invention is referred to as a Quadratic Residue Diffuser, or QRD.

N SECTION

29

23

19

17

13

11

7

31

...

97

0 1 4 9 16 25 36 49 64 81 3 24 47 72 2 31 62 95 33 70 12 53 96 44 91 43 94 50 8 65 27 88 54 22 89 61 35 11 86 66 48 32 18 6 93 85 79 75 73 73 75 79 85 93 6 18 32 48 66 86 11 35 61 89 22 54 88 27 65 8 50 94 43 91 44 96 53 12 70 33 95 62 31 2 72 47 24 3 81 64 49 36 25 16 9 4 1

0 1 4 2 2 4 1

0 1 4 9 5 3 3 5 9 4 1

0 1 4 9 3 12 10 10 12 3 9 4 1

0 1 4 9 16 8 2 15 13 13 15 2 8 16 9 4 1

0 1 4 9 16 6 17 11 7 5 5 7 11 17 6 16 9 4 1

0 1 4 9 16 2 13 3 18 12 8 6 6 8 12 18 3 13 2 16 9 4 1

0 1 4 9 16 25 7 20 6 23 13 5 28 24 22 22 24 28 5 13 23 6 20 7 25 16 9 4 1

0 1 4 9 16 25 5 18 2 19 7 28 20 14 10 8 8 10 14 20 28 7 19 2 18 5 25 16 9 4 1

Figure 3. Sections of well depths as N increases

Unlike earlier diffusive elements, Schroeder’s QRD made sound diffusion not only predictable, but also optimized. The under-

lying mathematics makes it possible to design a diffuser for a specific range of sound frequencies. The number sequence

dictating the panel design results in sound reflections scattered neatly across a wide angle and minimizes the chance of

unintended focal points. Because the wells are made with varying depths, sound will reflect off the individual bottoms at

slightly different times. So in addition to being scattered spatially, sound is also broken up temporarily.

Figure 5 contains the equations Schroeder used to design the Quadratic Residue Diffuser. N represents the number of wells

in the QRD design. Any number can be used for x and y to generate a quadratic number sequence, but QRD designs simply

use the number 1. QRD designs take into account the range of sound frequencies intended to be affected using frequen-

cy bounding equations to determine well dimensions. Well widths are dictated by the higher limit of sound frequencies to

affect; the higher the frequency limit, the narrower the well widths. Meanwhile, well depths are determined by the lower

frequency limit; the lower the frequency limit, the deeper the well.

Since its invention, QRD design has become standardized. Despite its parametric foundation, it is similarly constructed by

both commercial producers and DIY communities. In part, this is due to manufacturing limitations. As a result, QRD designs

commonly use smaller variables for N, with 7 being by far the most common. Larger numbers will increase the amount of

unique well depths and the complexity of construction.

Figure 4. Comparison of diffusion profiles

z = (x² * y²) mod Nz = x² mod N

ƒ = c / (4 * depth)ƒ = c / (2 * width)

c : 343.2 m/s ; 1126 ft/sƒ: frequency

λ = c / ƒ

Figure 5. Design equations for Schroeder’s diffuser

In most cases, when this diffuser is used in a space, identical panels are repeated along the length of the surface needing to

be treated. Unfortunately, because of the repetition, the resulting diffused sound may reflect in such a way that overlapping

occurs at some frequencies and along some angles, creating areas of noticeably higher and lower sound levels. Increasing

the distance between repeating elements can help with the matter. An aperiodic organization of the diffusers may also work,

but a single period device is preferred. Altogether, the issues that arise as a result of repetition can be avoided by increasing

the N-value in the quadratic equation. In doing this, we can generate a symmetrical number sequence of any length needed.

Using this method, we can stretch a single period over whatever surface we would like to mitigate and imbue a space with

an ideal diffusion profile.

Given that well widths are dictated by the upper limit of the frequency range we’d like to affect, we can use that number to

divide the length of the area we’d like to fill out with diffusive treatments. This number, rounded to the nearest prime number,

is used as the N-value of the quadratic design equation. After generating the proper quadratic number sequence, the largest

number should be associated to the deepest well, whose actual dimension is set using the frequency bounding equations.

All other numbers in between will dictate correlating well depths as a fractional function.

To reiterate, one reason this approach has not been done before has been the prohibitive cost of manufacturing the multi-

plicities required. But given the recent advances of machine manufacturing, the limitation could now be considered negligi-

ble. It is typically far more economical to make one hundred copies of one thing than to make one hundred unique versions.

With the use of computer automated fabrication techniques, the additional cost of doing the latter has been diminished

considerably. Even though the effect of a diffusive element is typically only noticed by the most trained of ears in the most

critical of listening environments, proper diffusion deployment should no longer be the result of fabrication limitations.

Figure 6. Scaled prototype

The fabrication method used to produce the prototypes could easily be adapted to produce the actual panels. After 3-di-

mensionally modeling the panels using custom algorithms based on the quadratic residue and frequency bounding equa-

tions, we extracted each unique profile along one direction of the panel. These profiles were then cut from some sheet

material with a thickness equal to the desired well width. After stacking all the profiles together in their proper order, the

panel is completed.

The initial intent of this research was to explore possibilities within acoustic design using a dynamic algorithmic modeling

approach. Our focus narrowed on diffusion and the design of corresponding architectural elements. Narrowing even more,

we developed a particular interest in Quadratic Residue Diffuser design and began extending its underlying mathemat-

ics. The process and understanding described above has been limited to this research effort. Beyond some rudimentary

ray-tracing tests, we have yet to validate the effectiveness of any prototype generated. Though this research started with

a focus on acoustic performance and followed into diffusion specifically, without any means to test our assumptions and

invention, we pivoted to a more unconstrained exploration into the relationship between modulus formulas and patterning.

In this way, divested of performance concerns, we applied the general techniques gained to a slightly more frenetic effort

generating patterns and evaluating them from an aesthetic stand point.

Figure 8. Scaled prototypeFigure 7. Screen capture

17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17

5^4 mod 32

0 29 8 15 0 9 24 27 0 21 8 7 0 1 24 19 0 13 8 31 0 25 24 11 0 5 8 23 0 17 24 3

5^3 mod 32

0 25 4 1 16 17 4 9 0 9 4 17 16 1 4 25 0 25 4 1 16 17 4 9 0 9 4 17 16 1 4 25

5^2 mod 32

5^1 mod 32

272217127229241914943126211611612823181383302520151050

4^4 mod 32

00000000000000000000000000000000

4^2 mod 32

160160160160160160160160160160160160160160160160

0 4 8 12 16 20 24 28 0 4 8 12 16 20 24 28 0 4 8 12 16 20 24 28 0 4 8 12 16 20 24 28

4^1 mod 32

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4^3 mod 32

2^6 mod 32

00000000000000000000000000000000

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2^5 mod 32

2^4 mod 32

160160160160160160160160160160160160160160160160

2^2 mod 32

4164041640416404164041640416404164041640

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

2^1 mod 32

0 8 0 24 0 8 0 24 0 8 0 24 0 8 0 24 0 8 0 24 0 8 0 24 0 8 0 24 0 8 0 24

2^3 mod 32

1^7 mod 32

3102101909070290270170150503025023013011010

1^5 mod 32

3101301102502305030170150290270907021019010

1^3 mod 32

31245038907241301181701524210198250232429027810

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

1^1 mod 32

0 1 4 9 16 25 4 17 0 17 4 25 16 9 4 1 0 1 4 9 16 25 4 17 0 17 4 25 16 9 4 1

1^2 mod 32

0 1 4 9 16 25 4 17 0 17 4 25 16 9 4 1 0 1 4 9 16 25 4 17 0 17 4 25 16 9 4 1

1^2 mod 32

0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1

1^4 mod 32

0 1 0 25 0 9 0 17 0 17 0 9 0 25 0 1 0 1 0 25 0 9 0 17 0 17 0 9 0 25 0 1

1^6 mod 32

0 15 30 13 28 11 26 9 24 7 22 5 20 3 18 1 16 31 14 29 12 27 10 25 8 23 6 21 4 19 2 17

15^1 mod 32

0 1 4 9 16 25 4 17 0 17 4 25 16 9 4 1 0 1 4 9 16 25 4 17 0 17 4 25 16 9 4 1

15^2 mod 32

0 15 24 21 0 19 8 25 0 23 24 29 0 27 8 1 0 31 24 5 0 3 8 9 0 7 24 13 0 11 8 17

15^3 mod 32

0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1

15^4 mod 32

0 13 26 7 20 1 14 27 8 21 2 15 28 9 22 3 16 29 10 23 4 17 30 11 24 5 18 31 12 25 6 19

13^1 mod 32

0 9 4 17 16 1 4 25 0 25 4 1 16 17 4 9 0 9 4 17 16 1 4 25 0 25 4 1 16 17 4 9

13^2 mod 32

3^4 mod 32

171610116170171610116170171610116170171610

0 21 8 23 0 1 24 3 0 13 8 15 0 25 24 27 0 5 8 7 0 17 24 19 0 29 8 31 0 9 24 11

13^3 mod 32

116170 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17

13^4 mod 32

0 19 24 1 0 7 8 21 0 27 24 9 0 15 8 29 0 3 24 17 0 23 8 5 0 11 24 25 0 31 8 13

11^3 mod 32

0 25 4 1 16 17 4 9 0 9 4 17 16 1 4 25 0 25 4 1 16 17 4 9 0 9 4 17 16 1 4 25

11^2 mod 323^2 mod 32

941716142502541161749094171614250254

0 11 22 1 12 23 2 13 24 3 14 25 4 15 26 5 16 27 6 17 28 7 18 29 8 19 30 9 20 31 10 21

11^1 mod 32

11617490

0 3 6 9 12 15 18 21 24 27 30 1 4 7 10 13 16 19 22 25 28 31 2 5 8 11

0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17 0 17 16 1 0 1 16 17

11^4 mod 32

14 17 20 23 26 29

3^1 mod 32

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8^4 mod 32

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8^3 mod 32

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8^2 mod 32

0 27 24 25 0 15 8 13 0 3 24 1 0 23 8 21 0 11 24 9 0 31 8 29

0 8 16 24 0 8 16 24 0 8 16 24 0 8 16 24 0 8 16 24 0 8 16 24 0 8 16 24 0 8 16 24

8^1 mod 32

0 19 24 17 0 7 8 5

3^3 mod 32

0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16

6^4 mod 32

0 24 0 8 0 24 0 8 0 24 0 8 0 24 0 8 0 24 0 8 0 24 0 8 0 24 0 8 0 24 0 8

6^3 mod 32

0 4 16 4 0 4 16 4 0 4 16 4 0 4 16 4 0 4 16 4 0 4 16 4 0 4 16 4 0 4 16 4

6^2 mod 32

0 19 0 9 0 15 0 5 0 11 0 1 0 7 0 29 0 3 0 25 0 31

0 6 12 18 24 30 4 10 16 22 28 2 8 14 20 26 0 6 12 18 24 30 4 10 16 22 28 2 8 14 20 26

6^1 mod 32

0 21 0 27 0 17 0 23 0 13

3^5 mod 32

7^1 mod 32

251811429221581261912530231692272013631241710328211470

7^2 mod 32

17425169410149162541701742516941014916254170

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7^3 mod 32

98190524150181102924702583021243101782701324230

0 0 0 0 0 0 0 0 0 0 0 0

16^4 mod 327^4 mod 32

116170171610116170171610116170171610116170171610

9^1 mod 32

231452819101241562920112251673021123261783122134271890

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9^2 mod 32

17425169410149162541701742516941014916254170

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

16^3 mod 32

0

9^3 mod 32

72429011810152450198902324130278170312421038250

9^4 mod 32

116170171610116170171610116170171610116170171610

0 0 0 0 0 0 0 0 0 0 0 0 0 0

10^1 mod 32

221222414426166281883020100221222414426166281883020100

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

16^2 mod 3210^2 mod 32

4164041640416404164041640416404164041640

10^3 mod 32

2408024080240802408024080240802408024080

0 16 0 16 0 16 0 16 0 16 0

10^4 mod 32

160160160160160160160160160160160160160160160160

16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16

16^1 mod 3212^1 mod 32

2082816424120208281642412020828164241202082816424120

12^2 mod 32

160160160160160160160160160160160160160160160160

12^3 mod 32

00000000000000000000000000000000

0 16 0 16 0 16 0 16 0

12^4 mod 32

00000000000000000000000000000000 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16 0 16

14^4 mod 32

14^1 mod 32

184228261230162206241028140184228261230162206241028140

14^2 mod 32

4164041640416404164041640416404164041640

14^3 mod 32

8024080240802408024080240802408024080240

1^2 mod 256

149162536496481100121144169196225033681051441852281764113164217167313219306513220116891642416414522857144233681610971964114424910020964177361531613741290129413716153361776420910024914441196970161682331445722814564241164891620113265019313273162171641136417228185144105683302251961691441211008164493625169410149162536496481100121144169196225033681051441852281764113164217167313219306513220116891642416414522857144233681610971964114424910020964177361531613741290129413716153361776420910024914441196970161682331445722814564241164891620113265019313273162171641136417228185144105683302251961691441211008164493625169410

1^2 mod 64

14916253649017365716414330334411657361704936251694101491625364901736571641433033441165736170493625169410

1^2 mod 16

1490941014909410

1^2 mod 4

1010

1^2 mod 1

0 0 1

1^2 mod 2

0 1 4 1 0 1 4 1

1^2 mod 8

0 1 4 9 16 25 4 17 0 17 4 25 16 9 4 1 0 1 4 9 16 25 4 17 0 17 4 25 16 9 4 1

1^2 mod 32

014916253649648110012116416897033681051657100176411336891673465065473168936113641710057161056833097684116121100816449362516941014916253649648110012116416897033681051657100176411336891673465065473168936113641710057161056833097684116121100816449362516941

1^2 mod 128