title: Computing Mod with a Variable Lookup Table

authors: Will, Mark Antony

subjects: Science & Technology

abstract: Encryption algorithms are designed to be difficult to break without knowledge of the secrets or keys. To achieve this, the algorithms require the keys to be large, with some having a recommend size of 2048-bits or more. However most modern processors only support computation on 64-bits at a time. Therefore standard operations with large numbers are more complicated to implement. One operation that is particularly challenging to efficiently implement is modular reduction. In this paper we propose a highly-efficient algorithm for solving large modulo operations; it has several advantages over current approaches as it supports the use of a variable sized lookup table, has good spatial and temporal locality allowing data to be streamed, and only requires basic processor instructions. Our proposed algorithm is theoretically compared to widely used modular algorithms, and shows improvements over other algorithms using predefined lookup tables.

text: Computing Mod with a Variable Lookup Table Encryption algorithms are designed to be difficult to break without knowledge of the secrets or keys. To achieve this, the algorithms require the keys to be large, with some having a recommend size of 2048-bits or more. However most modern processors only support computation on 64-bits at a time. Therefore standard operations with large numbers are more complicated to implement. One operation that is particularly challenging to efficiently implement is modular reduction. In this paper we propose a highly-efficient algorithm for solving large modulo operations; it has several advantages over current approaches as it supports the use of a variable sized lookup table, has good spatial and temporal locality allowing data to be streamed, and only requires basic processor instructions. Our proposed algorithm is theoretically compared to widely used modular algorithms, and shows improvements over other algorithms using predefined lookup tables.

year: 2016

title: Construction of Good Rank-1 Lattice Rules Based on the Weighted Star Discrepancy

authors: Joe, Stephen

subjects: mathematics

abstract: The ‘goodness’ of a set of quadrature points in [0, 1]d may be measured by the weighted star discrepancy. If the weights for the weighted star discrepancy are summable, then we show that for n prime there exist n-point rank-1 lattice rules whose weighted star discrepancy is O(n−1+δ) for any δ>0, where the implied constant depends on δ and the weights, but is independent of d and n. Further, we show that the generating vector z for such lattice rules may be obtained using a component-by-component construction. The results given here for the weighted star discrepancy are used to derive corresponding results for a weighted Lp discrepancy.

text: Construction of Good Rank-1 Lattice Rules Based on the Weighted Star Discrepancy The ‘goodness’ of a set of quadrature points in [0, 1]d may be measured by the weighted star discrepancy. If the weights for the weighted star discrepancy are summable, then we show that for n prime there exist n-point rank-1 lattice rules whose weighted star discrepancy is O(n−1+δ) for any δ>0, where the implied constant depends on δ and the weights, but is independent of d and n. Further, we show that the generating vector z for such lattice rules may be obtained using a component-by-component construction. The results given here for the weighted star discrepancy are used to derive corresponding results for a weighted Lp discrepancy.

year: 2006

title: Ko Te Kawa Tūpanapana i ngā Hau Tūpua a Tāwhiri-mātea : The validation, revitalisation and enhancement of Māori environment knowledge of weather and climate

authors: Skipper, Apanui Sonny

subjects: Maori Environmental knowledge

abstract: KO TE KAWA TŪPANAPANA I NGĀ HAU TŪPUA A TĀWHIRI-MĀTEA – THE RITUALISTIC FORMS GOVERNING THE SUPERNATURAL WINDS OF TĀWHIRI-MĀTEA
Mankind has always gazed skyward observing the im immense power generated by forked lightning when it strikes the land. The resounding thunder booming across the landscape never failed to capture the awe and imagination of a people. Māori were no different. The legacy of their Polynesian ancestors, who populated most of the Islands of Te Moana-nui-a-Kiwa, the largest ocean on the planet, based on belief, courage, and navigational expertise, honed over many millenia, reached Aotearoa approximately in the 12 century. These highly skilled ancestors from the tropics applied a formalised process of settling a new land. It also required their own weatherlore to be recalibrated to the local conditions, in order to survive in a much harsher, colder climate.

Over the centuries the descendants of these early Polynesian settlers, developed an extensive localised knowledge of weather, climate, and seasonal indicators to accurately predict daily weather, monthly or annual climate conditions. The ongoing lessons learned especially after experiencing a fatality were incorporated into cultural practices of important everyday activities such as ahuwhenua, te mahi hī ika, kohi rongoā, whakaako tamariki, gathering pounamu, whakatū rāhui, and kaitiakitanga, in order to minimise further loss of life.

Formal recognition of these contributions was made at the inaugural Māori Climate Forum in Wellington in 2003, where a number of Māori elders highlighted the importance of giving a greater account of Māori knowledge of environmental change. Among the many priorities identified for research at this forum, an opportunity based on a challenge from a Māori elder led to the development of a pilot project to explore traditional Māori understandings of weather–climate variability and change. 

One of the key outcomes from this work was the identification of Māori environmental indicators to anticipate local weather and climate conditions and thereby manage the risk associated with weather-climate extremes. However, in spite of the work achieved, there remain unanswered questions about the use and ongoing efficacy of environmental indicators to forecast and monitor weather and climate risks, including how new opportunities might be created to promote learning about subtle signals in nature that can reveal much about changes in weather and climate conditions. 

Due to the fact that many of the elders that provided the basis of this thesis have passed away, a great sense of responsibility has remained with me to complete this research. To identify the gaps in the previous research, to ground-truth it, to consolidate and carefully document the level of localised environmental knowledge of weather and climate specifically with iwi from Tainui-Hauraki, Te Whānau-a-Apanui, and Ngāi Tahu. A total of 56 interviews and 17 group wānanga were conducted during 2009-2016. Examples of Māori Environmental Knowledge were also sourced from numerous personal conversations with knowledge holders from 1989-2003, adding to the body of knowledge that was analysed to provide further context. A conceptual framework coined `Ngā Whenu Tapu e Waru’ was created from identifying the key themes of the interviews to inform how Māori Environmental Knowledge was described in the three regions. This study has yielded understandings of wind, rain, thunder, lightning, and cloud classifications; weather and climate indicators; spiritual signs of misfortune and death; memory techniques; longitudinal weather predictions, and lunar calendars in each region.

This thesis is underpinned by a Kaupapa Māori theory and research methodology that links to a body of transformative research and literature conducted by Māori scholars based on the knowledge and experience of tūpuna  Māori. 

Heoti anō rā kei ngā kākahi whakairoiro, kei ngā matataiko, nau piki mai, nau kake mai ki tēnei wānanga nui whakaharahara.

text: Ko Te Kawa Tūpanapana i ngā Hau Tūpua a Tāwhiri-mātea : The validation, revitalisation and enhancement of Māori environment knowledge of weather and climate KO TE KAWA TŪPANAPANA I NGĀ HAU TŪPUA A TĀWHIRI-MĀTEA – THE RITUALISTIC FORMS GOVERNING THE SUPERNATURAL WINDS OF TĀWHIRI-MĀTEA
Mankind has always gazed skyward observing the im immense power generated by forked lightning when it strikes the land. The resounding thunder booming across the landscape never failed to capture the awe and imagination of a people. Māori were no different. The legacy of their Polynesian ancestors, who populated most of the Islands of Te Moana-nui-a-Kiwa, the largest ocean on the planet, based on belief, courage, and navigational expertise, honed over many millenia, reached Aotearoa approximately in the 12 century. These highly skilled ancestors from the tropics applied a formalised process of settling a new land. It also required their own weatherlore to be recalibrated to the local conditions, in order to survive in a much harsher, colder climate.

Over the centuries the descendants of these early Polynesian settlers, developed an extensive localised knowledge of weather, climate, and seasonal indicators to accurately predict daily weather, monthly or annual climate conditions. The ongoing lessons learned especially after experiencing a fatality were incorporated into cultural practices of important everyday activities such as ahuwhenua, te mahi hī ika, kohi rongoā, whakaako tamariki, gathering pounamu, whakatū rāhui, and kaitiakitanga, in order to minimise further loss of life.

Formal recognition of these contributions was made at the inaugural Māori Climate Forum in Wellington in 2003, where a number of Māori elders highlighted the importance of giving a greater account of Māori knowledge of environmental change. Among the many priorities identified for research at this forum, an opportunity based on a challenge from a Māori elder led to the development of a pilot project to explore traditional Māori understandings of weather–climate variability and change. 

One of the key outcomes from this work was the identification of Māori environmental indicators to anticipate local weather and climate conditions and thereby manage the risk associated with weather-climate extremes. However, in spite of the work achieved, there remain unanswered questions about the use and ongoing efficacy of environmental indicators to forecast and monitor weather and climate risks, including how new opportunities might be created to promote learning about subtle signals in nature that can reveal much about changes in weather and climate conditions. 

Due to the fact that many of the elders that provided the basis of this thesis have passed away, a great sense of responsibility has remained with me to complete this research. To identify the gaps in the previous research, to ground-truth it, to consolidate and carefully document the level of localised environmental knowledge of weather and climate specifically with iwi from Tainui-Hauraki, Te Whānau-a-Apanui, and Ngāi Tahu. A total of 56 interviews and 17 group wānanga were conducted during 2009-2016. Examples of Māori Environmental Knowledge were also sourced from numerous personal conversations with knowledge holders from 1989-2003, adding to the body of knowledge that was analysed to provide further context. A conceptual framework coined `Ngā Whenu Tapu e Waru’ was created from identifying the key themes of the interviews to inform how Māori Environmental Knowledge was described in the three regions. This study has yielded understandings of wind, rain, thunder, lightning, and cloud classifications; weather and climate indicators; spiritual signs of misfortune and death; memory techniques; longitudinal weather predictions, and lunar calendars in each region.

This thesis is underpinned by a Kaupapa Māori theory and research methodology that links to a body of transformative research and literature conducted by Māori scholars based on the knowledge and experience of tūpuna  Māori. 

Heoti anō rā kei ngā kākahi whakairoiro, kei ngā matataiko, nau piki mai, nau kake mai ki tēnei wānanga nui whakaharahara.

year: 2020