Thursday, November 4, 2010
Sunday, October 17, 2010
K-LINK SALT
Come n join with us .... Bagi yang ingin mengetahui lebih jauh tentang K-LINK hubungi :
1 . PASKALIS AGUSTINUS , K-LINK JKID009636 , JAKARTA .
2 . ALBERTUS YONO HARDIN , K-LINK EID1209033 , boleh juga SMS di +6281339233834 , MANGGARAI - FLORES NTT .
3 . ANDRE OTTA DARKNET , K-LINK EID1204390 bisa PM or email di a.otta@yahoo.co.id , boleh juga SMS di +6285221111889 , +6285739129100 . Jalan anggrek no 17 oepura kupang NTT .
4 . MEDI LASIANGI , +6282144269720 , oesao kupang NTT .
5 . YOLANDA LASIANGI , +6285253061665 , oesao kupang NTT .
6 . YENI LASIANGI , +6285253061665 , oesao kupang NTT .
7 . IBU ROTE DENGKA , +6281339001754 , oesao kupang NTT .
8 . PETRUS KAPITAN , +6285239239952 , oesao kupang NTT .
9 . ROBERTUS .O. SIKONE , K-LINK IDNRAFA00017 , boleh juga SMS di +6285236461095 , 085253150998 ATAMBUA NTT .
10. AZEEZ BIN BRUNEI DARUSALAM , SMS di +6285237774588 , oesao kupang NTT .
11. NASIO LEMA , di +623808123934 penfui kupang NTT .
12 .TUAN GURU PANCOR , boleh juga SMS di +6281353604132 PRAYA BARAT LOMBOK NTB .
13. ASBEL BOLA boleh juga SMS / TLP di +6281331051363 ROTE NTT
14. JOSE 082144238223 .
15 . LEO ASHARI 081237960280 / 081353816606 .
16. RONI PAREIRA 081339586551 NAIKOLAN KUPANG NTT .
17 . LADISLAUS AMATUS JAKARTA .
WWW.OESAO.MULTIPLY.COM
כד יְבָרֶכְךָ יְהוָה, וְיִשְׁמְרֶךָ. The LORD bless thee, and keep thee;
כה יָאֵר יְהוָה פָּנָיו אֵלֶיךָ, וִיחֻנֶּךָּ. The LORD make His face to shine upon thee, and be gracious unto thee
כו יִשָּׂא יְהוָה פָּנָיו אֵלֶיךָ, וְיָשֵׂם לְךָ שָׁלוֹם. The LORD lift up His countenance upon thee, and give thee peace.
כז וְשָׂמוּ אֶת-שְׁמִי, עַל-בְּנֵי יִשְׂרָאֵל; וַאֲנִי, אֲבָרְכֵם. So shall they put My name upon the children of Israel, and I will bless them.
1 . PASKALIS AGUSTINUS , K-LINK JKID009636 , JAKARTA .
2 . ALBERTUS YONO HARDIN , K-LINK EID1209033 , boleh juga SMS di +6281339233834 , MANGGARAI - FLORES NTT .
3 . ANDRE OTTA DARKNET , K-LINK EID1204390 bisa PM or email di a.otta@yahoo.co.id , boleh juga SMS di +6285221111889 , +6285739129100 . Jalan anggrek no 17 oepura kupang NTT .
4 . MEDI LASIANGI , +6282144269720 , oesao kupang NTT .
5 . YOLANDA LASIANGI , +6285253061665 , oesao kupang NTT .
6 . YENI LASIANGI , +6285253061665 , oesao kupang NTT .
7 . IBU ROTE DENGKA , +6281339001754 , oesao kupang NTT .
8 . PETRUS KAPITAN , +6285239239952 , oesao kupang NTT .
9 . ROBERTUS .O. SIKONE , K-LINK IDNRAFA00017 , boleh juga SMS di +6285236461095 , 085253150998 ATAMBUA NTT .
10. AZEEZ BIN BRUNEI DARUSALAM , SMS di +6285237774588 , oesao kupang NTT .
11. NASIO LEMA , di +623808123934 penfui kupang NTT .
12 .TUAN GURU PANCOR , boleh juga SMS di +6281353604132 PRAYA BARAT LOMBOK NTB .
13. ASBEL BOLA boleh juga SMS / TLP di +6281331051363 ROTE NTT
14. JOSE 082144238223 .
15 . LEO ASHARI 081237960280 / 081353816606 .
16. RONI PAREIRA 081339586551 NAIKOLAN KUPANG NTT .
17 . LADISLAUS AMATUS JAKARTA .
WWW.OESAO.MULTIPLY.COM
כד יְבָרֶכְךָ יְהוָה, וְיִשְׁמְרֶךָ. The LORD bless thee, and keep thee;
כה יָאֵר יְהוָה פָּנָיו אֵלֶיךָ, וִיחֻנֶּךָּ. The LORD make His face to shine upon thee, and be gracious unto thee
כו יִשָּׂא יְהוָה פָּנָיו אֵלֶיךָ, וְיָשֵׂם לְךָ שָׁלוֹם. The LORD lift up His countenance upon thee, and give thee peace.
כז וְשָׂמוּ אֶת-שְׁמִי, עַל-בְּנֵי יִשְׂרָאֵל; וַאֲנִי, אֲבָרְכֵם. So shall they put My name upon the children of Israel, and I will bless them.
Thursday, October 14, 2010
Wednesday, October 6, 2010
Wednesday, September 29, 2010
Friday, September 24, 2010
obituary
Keluarga I Wayan Kari Pemilik Grup Waka. Basis operasional di Bali. Bidang usaha pariwisata (Waka Land Cruise, Waka di Ume, Waka Nusa, Waka Maya, Waka Gangga, Waka Shorea, Waka di Abian, Waka Namya, Waka Barong, Hotel Oberoi Bali dan Lombok, Waka Dive), konsultan manajemen, arsitektur (Sain D Sain), transportasi (taksi, rental mobil, kapal penumpang sepat), periklanan (Matamera Advertising), dan perumahan (menggarap hotel satu grup dan knockdown house), tiga hotel di Manado, dan satu di Bintan.
Anak Agung Ngurah Mahendra Basis operasional di Bali. Mendirikan PT Khrisna Kreasi pada 1985 di bidang usaha produksi dan eksportir garmen, forwarder dengan tiga cabang (Ubud, Jakarta, dan Surabaya), periklanan, perdagangan, jasa gudang, money changer, teknologi informasi, dan agen wisata. Perusahaannya kini berjumlah 12.
I Gde Wiratha dan Kadek Wiranatha Basis operasional di Bali. Mendirikan PT Gde & Kadek Brothers. Bidang usaha pariwisata (penginapan, biro perjalanan, restoran kafe, kapal pesiar, dan penerbangan). Group Bounty (Bounty Hotel, Hotel Barong, Dewi Sri Cottages, Vila Rumah Manis, Bounty Cruises, Paddy’s Cafe, Sari Club, Bounty Mall, Double Six, Gado Gado Restaurant, AJ Hackett Bungy, taksi Pan Witri dan Praja Taksi), biro perjalanan Calvin Tour & Travel, Bali Safari Rafting, Air Paradise International. Berencana membangun kembali Sari Club –yang dibom teroris pada 2002- dan sirkuit balap F1.
Gde Sumarjaya Linggih Pemilik Grup Ganeca. Basis operasional di Bali dan Bandung. Bidang usaha hotel (Hotel Sol Lovina berkapastas 120 kamar, 8 villa, dan satu president suite, serta hotel di Nusa Dua), printing supplier, dan minuman anggur (Indico Wine). Kini Grup Ganeca Prima membawahkan 11 anak usaha dan membangun Bali Trade Centre.
ABG Satria Naradha Pemilik Kelompok Media Bali Post. Basis operasional Bali dan Mataram. –tambahan dari saya, kini juga merambah Jakarta, Jogja, Bandung, Semarang, Palembang, dan Aceh-. Bidang usaha koran (Bali Post, Denpost, Bisnis Bali, Suara NTB, dan Prima), tabloid (Tokoh, Bali Travel News, Wiyata Mandala, dan Lintang), radio (Swara Widya Besakih, Global Kinijani, Genta Bali, Singaraja FM), TV (Bali TV, -ini tambahan dari saya: Jogja TV, Semarang TV, Bandung TV, Palembang TV, dan Aceh TV-). Pendapatan iklan Rp 198,3 milliar (Bali Post) dan Rp 33,3 milliar (Bali TV).
Putu Suryajaya Pemilik Grup Nikki yang berdiri pada 2000. Basis operasional di Denpasar dan Kuta. Bidang usaha hotel (Hotel Nikki), pusat kebugaran (Nikki Fitness Centre), rumah sakit (RS bersalin Puri Bunda), pusat pelatihan perhotelan berstandar internasional, dan waralaba sekolah Highscope. Saat ini juga mendirikan kondotel Nikki Denpasar dan kondotel lain di pinggiran Kuta.
AAM Sukadhana Wendha Pemilik Grup Kusemas yang berdiri pada 1990. Basis operasional di Denpasar. Bidang usaha di properti (spesialis pengembang rumah sederhana tipe 21 – 70 dan ruko), rumah mewah, migas, perbankan, travel, dan money changer. Sampai pertengahan 2006 telah membangun 8000 unit rumah sederhana di Bali (Permata Anyar, Griya Tantra Trisna, Bumi Dalung Permai, Permata Nambi, New Bumi Dalung Permai) dan 200 unit toko di Dalung. Ekspansi usaha di lima pompa bensin di Bali, kebun kelapa sawit 40 hektar di Kalimantan, rumah makan waralaba Ayam Bulungan, refleksi dan spa, gallery phone shop, bank perkreditan rakyat, minimarket, vila, resor, biro perjalanan, dan money changer.
Putu Agus Antara Pemilik PT Mama & Leon, hotel, dan properti. Bidang operasional di Denpasar. Bidang usaha garmen (PT Mama & Leon), hotel (hotel Keraton Jimbaran Resort), International Trade and Promotion Centre, Garuda Wisnu Kencana, dan The Renaissance. Ekspansi usaha merambah portofolio merk, pakaian kasual, kebaya, serta outlet.
Putu Subada Kusuma Basis operasional di Denpasar. Pemilik Hotel Sri Kusuma, workshop Melia Laundry, Toko Bunga Roses, dan biro hukum Putu Kusuma & Rekan. Pemilik master franchise 8 gerai waralaba Melia Laundry dan bermitra dengan petani mendirikan kebun bunga di Bedugul.
Keluarga Ida Bagus Tjetana Putra Pemilik Grup Santrian. Basis operasional di Denpasar. Bidang usaha perhotelan (Griya Santrian, Puri Santrian, Santrian Club), resto (Arena Sport Cafe, Mezzanine, dan The Village), transportasi (Sekar Menuh), rafting dan seawalker. Saat ini mendirikan villa seluas 2,5 hektar di Nusa Dua.
Nyoman Dana Asmara Pemilik CV Dana’s Company. Basis operasional di Denpasar. Bidang usaha rencana desain, desain, konstruksi hingga pemeliharaan produk dan eksportir rumah knock down dengan tarif antara US $100 – 600 ribu per proyek (biaya desain hingga pembangunan di luar transportasi, akomodasi, ongkos tukang, dan pengiriman material rumah dari Bali ke negara tujuan). Salah satu arsitek kelas dunia, anggota Asosiasi Arsitek Internasional. Karya: renovasi Bandara Ngurah Rai, Banyan Tree & Spa di Maladewa, rumah keluarga Raja Fahd Arab Saudi, dll. I
Gede Agus Hardiawan Pemilik PT Hardys Retailindo. Basis operasional di Bali dan Jawa Timur. Bidang usaha super market yang tersebar di seluruh kabupaten/kota di Bali dan beberapa kota lain seperti Banyuwangi, Jember, dan Mataram. Hardy’s kini bekerja sama dengan PT Ramayana Lestari Sentosa.
Djuwito Tjahjadi PT Putra Bhineka Perkasa, produsen kopi merk Kupu-kupu Bola Dunia. Bidang operasional di Denpasar. Berdiri sejak 1935 dan usaha di bidang pengolahan dan perdagangan kopi, termasuk semua hal yang berhubungan dengan kopi seperti pelatihan tentang kopi. Mendirikan Jazz Bar & Grill Cafe dan Kopi Bali House.
Desak Nyoman SuartiPemilik PT Suarti (Suarti Collection). Basis operasional di Gianyar sejak 1990. Bidang usaha perhiasan dan home wear dari perak murni. Punya gerai di Sanur, Ubud, dan Kuta serta satu toko di New York. Lebih dari 90 persen produk diekspor ke Inggris, Italia, Amerika Serikat, Jepang, dan Australia.
Gde Ngurah Wididana alias Pak Oles Pemilik PT Karya Pak Oles Tokcer. Bidang usaha produksi dan perdagangan obat alternatif, pupuk alternatif, resto, media (dua koran dan tiga radio), klinik pengobatan, dan lembaga penelitian dan pendidikan. Jumlah produk 32 buah muali dari madu, jamu, gelang penyembuh, pupuk, hingga penghemat bahan bakar kendaraan. Punya 39 kantor cabang pemasaran.
Joseph Theodorus Wulianadi alias Joger Pemilik CV Wira’s Garment Melania Soraya, produsen kaos Joger. Basis operasional di Kuta. Bidang usaha kaos dan merchandise Joger dan Jok Mah Li (pojok mahal sekali, barang-barang dari luar negeri dengan harga miring). Menjual sekitar 10 ribu item barang dengan marjin yang diambil 5,8 persen. Jumlah gerai hanya satu di Kuta tapi selalu penuh dengan wisatawan.
Bagus Sudibya Pemilik Bagus Discovery. Basis operasional di Bali. Bidang usaha pariwisata dan agrobisnis. Berdiri pada 1978 dengan Baruna Water Sport, perintis bisnis menyelam di Bali dan Pulau Komodo. Mendirikan Nusa Dua Tour & Travel, Puri Bagus Manggis, Puri Bagus Candidasa Villa Resort & Spa, Puri Bagus Lovina Villa Resort & Spa, Bagus Jati Health & Wellbeing Retreat, The Baliyem Valley Resort di Wamena Papua, Bagus Agro Pelaga (agrobisnis dan agrowisata), dan Bagus Agro Ponjok (pemasok bahan baku ke hotel satu grup dan supermarket di Bali).
Jaya Susila Pemilik Grup Alpha. Basis operasional di Denpasar di bidang kargo sejak 1978. Bidang usaha eksportir garmen dan kerajinan tangan (PT Alpha Sigma Bali), kargo (PT Alpha Sigma Cargo), konsultan bisnis dan pengadaan software (PT Sari Alpha Dwi Karya). Aset sekitar Rp 10 milyar.
Panudiana Kuhn Pemilik PT Dianatina Ayu dan PT Dianasurya Ratna Cargo. Basis operasional di Kuta. Bidang usaha garmen, kargo, dan penginapan. Berdiri pada 1985 (CV Diana dan pada 1993 menjadi PT Dianatina Ayu). Memproduksi garmen pesanan dari merk internasional seperti Bebob, Transparant, dan Tbob dengan pasar Eropa, Kanada, AS, dan Australia. Pemilik Vila Diana Bali, Hotel Ratna Bali, dan Istana Ratna Hotel Yogyakarta.
Tjok Oka Artha Ardhana S Pemilik Grup Tjampuhan. Basis operasional di Ubud. Bidang usaha puri, hotel, spa, dan sekolah. Meneruskan usaha keluarga pada 1984. Puri Tjampuhan mengembangkan usaha jadi 60 unit bungalow, mendirikan Puri Pita Maha, Hotel Royal Pita Maha, Kirana Spa, Sekolah Tinggi, dan berbagai yayasan.
M Sunhaji Pemilk Grup Risun. Basis operasional di Bali. Bidang usaha budi daya dan perdagangan mutiara lepas dan perhiasan mutiara (Risun Pearl) dan jasa pemasaran serta penyewaan properti (tanah, vila, dan rumah). Pemilik tiga gerai di Sogo, Discovery Shopping Mall, dan Risun Pearl di Kuta.
*
Anak Agung Ngurah Mahendra Basis operasional di Bali. Mendirikan PT Khrisna Kreasi pada 1985 di bidang usaha produksi dan eksportir garmen, forwarder dengan tiga cabang (Ubud, Jakarta, dan Surabaya), periklanan, perdagangan, jasa gudang, money changer, teknologi informasi, dan agen wisata. Perusahaannya kini berjumlah 12.
I Gde Wiratha dan Kadek Wiranatha Basis operasional di Bali. Mendirikan PT Gde & Kadek Brothers. Bidang usaha pariwisata (penginapan, biro perjalanan, restoran kafe, kapal pesiar, dan penerbangan). Group Bounty (Bounty Hotel, Hotel Barong, Dewi Sri Cottages, Vila Rumah Manis, Bounty Cruises, Paddy’s Cafe, Sari Club, Bounty Mall, Double Six, Gado Gado Restaurant, AJ Hackett Bungy, taksi Pan Witri dan Praja Taksi), biro perjalanan Calvin Tour & Travel, Bali Safari Rafting, Air Paradise International. Berencana membangun kembali Sari Club –yang dibom teroris pada 2002- dan sirkuit balap F1.
Gde Sumarjaya Linggih Pemilik Grup Ganeca. Basis operasional di Bali dan Bandung. Bidang usaha hotel (Hotel Sol Lovina berkapastas 120 kamar, 8 villa, dan satu president suite, serta hotel di Nusa Dua), printing supplier, dan minuman anggur (Indico Wine). Kini Grup Ganeca Prima membawahkan 11 anak usaha dan membangun Bali Trade Centre.
ABG Satria Naradha Pemilik Kelompok Media Bali Post. Basis operasional Bali dan Mataram. –tambahan dari saya, kini juga merambah Jakarta, Jogja, Bandung, Semarang, Palembang, dan Aceh-. Bidang usaha koran (Bali Post, Denpost, Bisnis Bali, Suara NTB, dan Prima), tabloid (Tokoh, Bali Travel News, Wiyata Mandala, dan Lintang), radio (Swara Widya Besakih, Global Kinijani, Genta Bali, Singaraja FM), TV (Bali TV, -ini tambahan dari saya: Jogja TV, Semarang TV, Bandung TV, Palembang TV, dan Aceh TV-). Pendapatan iklan Rp 198,3 milliar (Bali Post) dan Rp 33,3 milliar (Bali TV).
Putu Suryajaya Pemilik Grup Nikki yang berdiri pada 2000. Basis operasional di Denpasar dan Kuta. Bidang usaha hotel (Hotel Nikki), pusat kebugaran (Nikki Fitness Centre), rumah sakit (RS bersalin Puri Bunda), pusat pelatihan perhotelan berstandar internasional, dan waralaba sekolah Highscope. Saat ini juga mendirikan kondotel Nikki Denpasar dan kondotel lain di pinggiran Kuta.
AAM Sukadhana Wendha Pemilik Grup Kusemas yang berdiri pada 1990. Basis operasional di Denpasar. Bidang usaha di properti (spesialis pengembang rumah sederhana tipe 21 – 70 dan ruko), rumah mewah, migas, perbankan, travel, dan money changer. Sampai pertengahan 2006 telah membangun 8000 unit rumah sederhana di Bali (Permata Anyar, Griya Tantra Trisna, Bumi Dalung Permai, Permata Nambi, New Bumi Dalung Permai) dan 200 unit toko di Dalung. Ekspansi usaha di lima pompa bensin di Bali, kebun kelapa sawit 40 hektar di Kalimantan, rumah makan waralaba Ayam Bulungan, refleksi dan spa, gallery phone shop, bank perkreditan rakyat, minimarket, vila, resor, biro perjalanan, dan money changer.
Putu Agus Antara Pemilik PT Mama & Leon, hotel, dan properti. Bidang operasional di Denpasar. Bidang usaha garmen (PT Mama & Leon), hotel (hotel Keraton Jimbaran Resort), International Trade and Promotion Centre, Garuda Wisnu Kencana, dan The Renaissance. Ekspansi usaha merambah portofolio merk, pakaian kasual, kebaya, serta outlet.
Putu Subada Kusuma Basis operasional di Denpasar. Pemilik Hotel Sri Kusuma, workshop Melia Laundry, Toko Bunga Roses, dan biro hukum Putu Kusuma & Rekan. Pemilik master franchise 8 gerai waralaba Melia Laundry dan bermitra dengan petani mendirikan kebun bunga di Bedugul.
Keluarga Ida Bagus Tjetana Putra Pemilik Grup Santrian. Basis operasional di Denpasar. Bidang usaha perhotelan (Griya Santrian, Puri Santrian, Santrian Club), resto (Arena Sport Cafe, Mezzanine, dan The Village), transportasi (Sekar Menuh), rafting dan seawalker. Saat ini mendirikan villa seluas 2,5 hektar di Nusa Dua.
Nyoman Dana Asmara Pemilik CV Dana’s Company. Basis operasional di Denpasar. Bidang usaha rencana desain, desain, konstruksi hingga pemeliharaan produk dan eksportir rumah knock down dengan tarif antara US $100 – 600 ribu per proyek (biaya desain hingga pembangunan di luar transportasi, akomodasi, ongkos tukang, dan pengiriman material rumah dari Bali ke negara tujuan). Salah satu arsitek kelas dunia, anggota Asosiasi Arsitek Internasional. Karya: renovasi Bandara Ngurah Rai, Banyan Tree & Spa di Maladewa, rumah keluarga Raja Fahd Arab Saudi, dll. I
Gede Agus Hardiawan Pemilik PT Hardys Retailindo. Basis operasional di Bali dan Jawa Timur. Bidang usaha super market yang tersebar di seluruh kabupaten/kota di Bali dan beberapa kota lain seperti Banyuwangi, Jember, dan Mataram. Hardy’s kini bekerja sama dengan PT Ramayana Lestari Sentosa.
Djuwito Tjahjadi PT Putra Bhineka Perkasa, produsen kopi merk Kupu-kupu Bola Dunia. Bidang operasional di Denpasar. Berdiri sejak 1935 dan usaha di bidang pengolahan dan perdagangan kopi, termasuk semua hal yang berhubungan dengan kopi seperti pelatihan tentang kopi. Mendirikan Jazz Bar & Grill Cafe dan Kopi Bali House.
Desak Nyoman SuartiPemilik PT Suarti (Suarti Collection). Basis operasional di Gianyar sejak 1990. Bidang usaha perhiasan dan home wear dari perak murni. Punya gerai di Sanur, Ubud, dan Kuta serta satu toko di New York. Lebih dari 90 persen produk diekspor ke Inggris, Italia, Amerika Serikat, Jepang, dan Australia.
Gde Ngurah Wididana alias Pak Oles Pemilik PT Karya Pak Oles Tokcer. Bidang usaha produksi dan perdagangan obat alternatif, pupuk alternatif, resto, media (dua koran dan tiga radio), klinik pengobatan, dan lembaga penelitian dan pendidikan. Jumlah produk 32 buah muali dari madu, jamu, gelang penyembuh, pupuk, hingga penghemat bahan bakar kendaraan. Punya 39 kantor cabang pemasaran.
Joseph Theodorus Wulianadi alias Joger Pemilik CV Wira’s Garment Melania Soraya, produsen kaos Joger. Basis operasional di Kuta. Bidang usaha kaos dan merchandise Joger dan Jok Mah Li (pojok mahal sekali, barang-barang dari luar negeri dengan harga miring). Menjual sekitar 10 ribu item barang dengan marjin yang diambil 5,8 persen. Jumlah gerai hanya satu di Kuta tapi selalu penuh dengan wisatawan.
Bagus Sudibya Pemilik Bagus Discovery. Basis operasional di Bali. Bidang usaha pariwisata dan agrobisnis. Berdiri pada 1978 dengan Baruna Water Sport, perintis bisnis menyelam di Bali dan Pulau Komodo. Mendirikan Nusa Dua Tour & Travel, Puri Bagus Manggis, Puri Bagus Candidasa Villa Resort & Spa, Puri Bagus Lovina Villa Resort & Spa, Bagus Jati Health & Wellbeing Retreat, The Baliyem Valley Resort di Wamena Papua, Bagus Agro Pelaga (agrobisnis dan agrowisata), dan Bagus Agro Ponjok (pemasok bahan baku ke hotel satu grup dan supermarket di Bali).
Jaya Susila Pemilik Grup Alpha. Basis operasional di Denpasar di bidang kargo sejak 1978. Bidang usaha eksportir garmen dan kerajinan tangan (PT Alpha Sigma Bali), kargo (PT Alpha Sigma Cargo), konsultan bisnis dan pengadaan software (PT Sari Alpha Dwi Karya). Aset sekitar Rp 10 milyar.
Panudiana Kuhn Pemilik PT Dianatina Ayu dan PT Dianasurya Ratna Cargo. Basis operasional di Kuta. Bidang usaha garmen, kargo, dan penginapan. Berdiri pada 1985 (CV Diana dan pada 1993 menjadi PT Dianatina Ayu). Memproduksi garmen pesanan dari merk internasional seperti Bebob, Transparant, dan Tbob dengan pasar Eropa, Kanada, AS, dan Australia. Pemilik Vila Diana Bali, Hotel Ratna Bali, dan Istana Ratna Hotel Yogyakarta.
Tjok Oka Artha Ardhana S Pemilik Grup Tjampuhan. Basis operasional di Ubud. Bidang usaha puri, hotel, spa, dan sekolah. Meneruskan usaha keluarga pada 1984. Puri Tjampuhan mengembangkan usaha jadi 60 unit bungalow, mendirikan Puri Pita Maha, Hotel Royal Pita Maha, Kirana Spa, Sekolah Tinggi, dan berbagai yayasan.
M Sunhaji Pemilk Grup Risun. Basis operasional di Bali. Bidang usaha budi daya dan perdagangan mutiara lepas dan perhiasan mutiara (Risun Pearl) dan jasa pemasaran serta penyewaan properti (tanah, vila, dan rumah). Pemilik tiga gerai di Sogo, Discovery Shopping Mall, dan Risun Pearl di Kuta.
*
Product knowledge
K-Link Kino / Kinotakara
Cleansing : Pembersihan Toksin
Balancing : Penyeimbangan
kino kinitajara k-link kino, koyo kesehatan untuk membuang racun dalam tubuh
Manfaat K-Link Kino / Kinotakara :
* Mampu menyerap toksin / racun dalam tubuh
* Membantu meringankan rasa sakit pada penderita reumatik dan radang sendi.
* Membantu memperlancar sirkulasi darah
* Meningkatkan kualitas tidur
* Memperbaiki metabolisme tubuh
* Mengaktifkan meridian refleksi (titik akupuntur)
K-Link Riddance
Cleansing : Pembersihan Toksin
membersihkan usus, awet muda, hidup lebih sehat
Manfaat K-Link Riddance :
* Pembersihan dan melancarkan aliran darah
* Membersihkan usus dan memperbaiki sifat peristaltik usus
* Meningkatkan Daya detoksifikasi alami pada tubuh kita
* Penyeimbang fungsi organ
* Meningkatkan daya serap nutrisi
* Memperlambat proses penuaan
K-Link Liquid Chlorophyll
Cleansing : Pembersihan , Balancing : Penyeimbang
Activating : Pengaktifan
minuman berenergi , menyembuhkan luka , anti kolesterol
Manfaat K-Link Liquid Chlorophyll:
* Mengeluarkan toksin / racun dan zat kimia berbahaya melalui organ pencernaan.
* Menjaga keseimbangan kolesterol, gula darah, asam basa tubuh dan sistim hormonal.
* Membantu proses penyembuhan
* Menghalangi pertumbuhan bakteri dan mempercepat penyembuhan luka
* Penyeimbang fungsi organ
* Berfungsi sebagai anti kanker
* Nutrisi 1 Sendok Makan Chlorophyll = 1 Kg Sayuran
K-Link OmegaSqua
Cleansing : Pembersihan
Balancing : Penyeimbang
menyembuhkan penyakit jantung, hipertensi, kesehatan jantung, kesehatan kulit, omega 3 dengan DHA dan EPA tinggi
Manfaat K-Link OmegaSqua:
* Mencegah penyakit jantung koroner.
* Mencegah penyempitan pembuluh darah
* Membersihkan Plaque pada pembuluh darah
* Menurunkan kadar Trigliserida
* Menormalkan tekanan darah
* Meningkatkan elastisitas pembuluh darah
* Meningkatkan kadar oksigen dalam sel tubuh
* Menjaga kesehatan kulit dan meningkatkan kekebalan tubuh
K-Link Liquid Organic Spirulina
Balancing : Penyeimbangan
Activating : Pengaktifan
menurunkan stress, minuman bernutrisi tinggi, membantu proses diet
Manfaat K-Link Liquid Organic Spirulina:
* Memberikan kandungan nutrisi yang sangat lengkap
* Membantu menurunkan kadar kolesterol
* Berguna bagi penderita kencing manis dan hipertensi
* Berfungsi sebagai anti kanker (Beta Carotene)
* Meningkatkan daya tahan tubuh
* Mengurangi efek radiasi (kemoterapi)
* Membantu proses diet kesehatan
* Menurunkan kadar stress dan depresi
K-Link Gamat
Balancing : Penyeimbangan
Activating : Pengaktifan
menurunkan kolesterol, anti inflamasi, menyembuhkan luka, regenerasi sel
Manfaat K-Link Gamat:
* Menurunkan kadar kolesterol
* Anti inflamasi
* Membantu penyembuhan sinusitis (Beta Gamat Emulsion)
* Meningkatkan Kesuburan pria dan wanita (Beta Gamat Emulsion)
* Mempercepat penyembuhan luka (radang, Jerawat, Luka bakar)
* Meningkatkan stamina dan daya tahan tubuh
* Menghaluskan kulit (Gamat Vitagel)
* Menyembuhkan penyakit kulit (Gamat Vitagel)
K-Link Teh Rooibos SOD
Balancing : Penyeimbangan
Activating : Pengaktifan
menurunkan kolesterol, anti inflamasi, menyembuhkan luka, regenerasi sel
Manfaat K-Link Teh Rooibos SOD :
* Menguatkan tulang dan gigi
* Memperlancar metabolisme
* Meningkatkan oksigen dan memperlancar peredaran darah
* Meningkatkan Kesuburan pria dan wanita (Beta Gamat Emulsion)
* Menyegarkan system syaraf
* Meningkatkan kesehatan kulit
* Isi 40 sachet (1 sachet untuk 1,5 liter Air)
Cleansing : Pembersihan Toksin
Balancing : Penyeimbangan
kino kinitajara k-link kino, koyo kesehatan untuk membuang racun dalam tubuh
Manfaat K-Link Kino / Kinotakara :
* Mampu menyerap toksin / racun dalam tubuh
* Membantu meringankan rasa sakit pada penderita reumatik dan radang sendi.
* Membantu memperlancar sirkulasi darah
* Meningkatkan kualitas tidur
* Memperbaiki metabolisme tubuh
* Mengaktifkan meridian refleksi (titik akupuntur)
K-Link Riddance
Cleansing : Pembersihan Toksin
membersihkan usus, awet muda, hidup lebih sehat
Manfaat K-Link Riddance :
* Pembersihan dan melancarkan aliran darah
* Membersihkan usus dan memperbaiki sifat peristaltik usus
* Meningkatkan Daya detoksifikasi alami pada tubuh kita
* Penyeimbang fungsi organ
* Meningkatkan daya serap nutrisi
* Memperlambat proses penuaan
K-Link Liquid Chlorophyll
Cleansing : Pembersihan , Balancing : Penyeimbang
Activating : Pengaktifan
minuman berenergi , menyembuhkan luka , anti kolesterol
Manfaat K-Link Liquid Chlorophyll:
* Mengeluarkan toksin / racun dan zat kimia berbahaya melalui organ pencernaan.
* Menjaga keseimbangan kolesterol, gula darah, asam basa tubuh dan sistim hormonal.
* Membantu proses penyembuhan
* Menghalangi pertumbuhan bakteri dan mempercepat penyembuhan luka
* Penyeimbang fungsi organ
* Berfungsi sebagai anti kanker
* Nutrisi 1 Sendok Makan Chlorophyll = 1 Kg Sayuran
K-Link OmegaSqua
Cleansing : Pembersihan
Balancing : Penyeimbang
menyembuhkan penyakit jantung, hipertensi, kesehatan jantung, kesehatan kulit, omega 3 dengan DHA dan EPA tinggi
Manfaat K-Link OmegaSqua:
* Mencegah penyakit jantung koroner.
* Mencegah penyempitan pembuluh darah
* Membersihkan Plaque pada pembuluh darah
* Menurunkan kadar Trigliserida
* Menormalkan tekanan darah
* Meningkatkan elastisitas pembuluh darah
* Meningkatkan kadar oksigen dalam sel tubuh
* Menjaga kesehatan kulit dan meningkatkan kekebalan tubuh
K-Link Liquid Organic Spirulina
Balancing : Penyeimbangan
Activating : Pengaktifan
menurunkan stress, minuman bernutrisi tinggi, membantu proses diet
Manfaat K-Link Liquid Organic Spirulina:
* Memberikan kandungan nutrisi yang sangat lengkap
* Membantu menurunkan kadar kolesterol
* Berguna bagi penderita kencing manis dan hipertensi
* Berfungsi sebagai anti kanker (Beta Carotene)
* Meningkatkan daya tahan tubuh
* Mengurangi efek radiasi (kemoterapi)
* Membantu proses diet kesehatan
* Menurunkan kadar stress dan depresi
K-Link Gamat
Balancing : Penyeimbangan
Activating : Pengaktifan
menurunkan kolesterol, anti inflamasi, menyembuhkan luka, regenerasi sel
Manfaat K-Link Gamat:
* Menurunkan kadar kolesterol
* Anti inflamasi
* Membantu penyembuhan sinusitis (Beta Gamat Emulsion)
* Meningkatkan Kesuburan pria dan wanita (Beta Gamat Emulsion)
* Mempercepat penyembuhan luka (radang, Jerawat, Luka bakar)
* Meningkatkan stamina dan daya tahan tubuh
* Menghaluskan kulit (Gamat Vitagel)
* Menyembuhkan penyakit kulit (Gamat Vitagel)
K-Link Teh Rooibos SOD
Balancing : Penyeimbangan
Activating : Pengaktifan
menurunkan kolesterol, anti inflamasi, menyembuhkan luka, regenerasi sel
Manfaat K-Link Teh Rooibos SOD :
* Menguatkan tulang dan gigi
* Memperlancar metabolisme
* Meningkatkan oksigen dan memperlancar peredaran darah
* Meningkatkan Kesuburan pria dan wanita (Beta Gamat Emulsion)
* Menyegarkan system syaraf
* Meningkatkan kesehatan kulit
* Isi 40 sachet (1 sachet untuk 1,5 liter Air)
Thursday, September 23, 2010
Science & Technology
The 100th smallest country, with less than 1/1000th of the
world's population, can lay claim to the following:
The cell phone was first developed at the Motorola plant in
Israel.
Most of the Windows NT and XP operating systems were
developed by Microsoft-Israel.
The Pentium MMX Chip technology was designed in Israel at
Intel.
Both the Pentium-4 microprocessor for desktop computers and
the Centrino processor for laptops were entirely designed,
developed and produced in Israel.
Voice mail technology was developed in Israel. The Israeli
company Amdocs is the largest company in the world in this
field.
Both Microsoft and Cisco built their only foreign-based
research and development facilities in Israel.
The program ICQ, which is the technological basis for AOL
Instant Messenger, was developed in 1996 by four young
Israelis.
Disk on Key - a portable, virtual hard disk - was developed
by the Israeli company M-Systems.
Israel has the highest number of personal computers per
capita in the world.
Israel has the highest number of university degrees per
capita in the world.
Israel produces more scientific papers per capita than any
other nation by a large margin - 109 per 10,000 people - as
well as one of the highest per capita rates of patents
filed.
In proportion to its population, Israel has the largest
number of startup companies in the world. In absolute terms,
Israel has the largest number of startup companies than any
other country in the world, except the US.
With more than 3,000 high-tech companies and startups,
Israel has the highest concentration of hi-tech companies in
the world - apart from Silicon Valley.
Israel is ranked #2 in the world for venture capital funds
right behind the United States.
Outside the United States and Canada, Israel has the largest
number of companies listed on NASDAQ.
Israel has the highest average living standards in the
Middle East. The per capita income in 2000 was over $17,500,
exceeding that of the United Kingdom.
On a per capita basis, Israel has the largest number of
biotech startups.
Twenty four percent of Israel's workforce holds university
degrees - ranking third in the industrialized world, after
the United States and Holland - and 12 percent hold advanced
degrees.
Israel has the third highest rate of entrepreneurship - and
the highest rate among women and among people over 55 in the
world.
Relative to its population, Israel is the largest
immigrant-absorbing nation on earth.
Israel has the world's second highest supply of new books
per capita.
Israel has more museums per capita than any other country.
Israeli scientists developed the first fully computerized,
no-radiation diagnostic instrumentation for breast cancer.
An Israeli company developed a computerized system for
ensuring proper administration of medications, thus removing
human error from medical treatment. Every year in U. S.
hospitals 7,000 patients die from treatment mistakes.
Israel's Given Imaging developed the PillCam - the first
ingestible video camera, which is so small it fits inside a
pill. Used to view the small intestine from the inside, the
camera helps doctors diagnose cancer and digestive
disorders.
Researchers in Israel developed a new device that directly
helps the heart pump blood. The new device is synchronized
with the heart's mechanical operations through a
sophisticated system of sensors.
Israel leads the world in the number of scientists and
technicians in the workforce, with 145 per 10,000, as
opposed to 85 in the U.S., over 70 in Japan, and less than
60 in Germany.
A new acne treatment developed in Israel causes acne
bacteria to self-destruct - all without damaging
surroundings skin or tissue.
An Israeli company was the first to develop and install a
large-scale solar-powered and fully functional electricity
generating plant in Southern California's Mojave Desert.
The first computer anti-virus software package was developed
in Israel back in the 1970's.
Major law enforcement agencies use Israeli technologies to
monitor voices and messages on conventional phones, mobile
phones and e-mails.
An Israeli company, Teva, is the world's largest generic
pharmaceutical company.
A new brain implant has been developed in Israel that can
lower the risk of stroke by diverting blood clots away from
sensitive areas of the brain.
IBM scientists in Israel are playing a vital role in a
massive project of the European Organization for Nuclear
Research (CERN) to discover the origins of life on earth.
Israeli software company Check Point is the global leader in
Virtual Private Network (VPN) and firewall technologies.
Israeli company Elta is responsible for the world's first
civilian aircraft equipped with technology designed to
protect airliners from a missile attack.
Mashav, the Israeli Foreign Ministry's Center for
International Cooperation has trained over 200,000
international aid workers that have traveled to dozens of
countries to help with medicine, agriculture, disaster
relief, and many other issues.
Israel has, for many years, held the world record in milk
production.
Rummikub, the third highest selling board game in the world,
is manufactured in a family-run plant in the small southern
Israeli town of Arad.
Drip irrigation - the system that is based on using plastic
pipes that release small amounts of water next to crops or
plants - was developed by the Israeli engineer Simcha Blas
in the 1970's. The invention caused a revolution in
agriculture.
A design submitted by Israeli-born Michael Arad has been
chosen for the World Trade Center Memorial, from amongst
5,000 entries from around the world.
Israeli company Retalix created the grocery scanners used at
such stores as Costco, Albertson's, and 7-11, as well as
25,000 additional stores and quick-service restaurants
throughout the United States.
Primate research at Hebrew University is leading to the
development of a robotic arm that can respond to the brain
commands of a paralyzed person.
Two Israeli researchers are generating cancer-killing
molecules that will recognize cancerous cells and target
them aggressively, while not affecting normal cells.
Israeli researchers developed a novel stem cell therapy to
treat Parkinson's Disease - using a patient's own bone
marrow stem cells to produce the missing chemical that
enables restoration of motor movement.
Israeli company Silent Communications has developed a type
of silent conversation system for cell phones, so users can
carry on conversations without saying a word.
The Israeli company Wondernet is currently dominating the
world market in document signature authentication, with its
unique scientific method of verifying handwritten
signatures.
Israeli Professor Yehuda Finkelstein has discovered the
cause of and cure for halitosis (bad breath).
Cherry tomatoes were originally supposed to be a snack when
they were designed by a group of scientists led by professor
Nahum Keidar from the agriculture faculty at the Weizmann
Institute of Science, with the cooperation of the Israeli
company Zera.
The Quicktionary, a pen size scanner that scans a word or a
sentence and translates it to a different language, was
developed by the Wizcom Company, based in Jerusalem.
Professor Ehud Keinan from the Technion Israel Institute of
Technology developed a pen that identifies an improvised
explosive.
The Israeli company Insightec developed an ultrasound system
for removing tumors without surgery.
Researchers at the Technion have developed an antibiotic
that destroys anthrax bacteria as well as the toxins it
secretes into the bloodstream of the infected body.
Epilady, an electric hair removal system, was developed by
Yair Dar and Shimon Yahav from the Goshrim Kibbutz.
The sun-heated water tank, a device that converts solar
energy into thermal energy and that saves about 4% of the
national energy supply, was developed by an engineer from
Jerusalem.
Dr. Gal Yadid, Dr. Rachel Mayan, and Professor Abraham
Weizman from Bar Ilan University developed a form of drug
rehabilitation using a natural steroid that is inserted into
the brain and develops a resistance for the drugs.
Alon Moses from Hadassah Medical Center in Jerusalem and
Imanuel Hensky and Carlos Hidelgo-Grass from Hebrew
University decoded the mechanism for Streptococcus A.
world's population, can lay claim to the following:
The cell phone was first developed at the Motorola plant in
Israel.
Most of the Windows NT and XP operating systems were
developed by Microsoft-Israel.
The Pentium MMX Chip technology was designed in Israel at
Intel.
Both the Pentium-4 microprocessor for desktop computers and
the Centrino processor for laptops were entirely designed,
developed and produced in Israel.
Voice mail technology was developed in Israel. The Israeli
company Amdocs is the largest company in the world in this
field.
Both Microsoft and Cisco built their only foreign-based
research and development facilities in Israel.
The program ICQ, which is the technological basis for AOL
Instant Messenger, was developed in 1996 by four young
Israelis.
Disk on Key - a portable, virtual hard disk - was developed
by the Israeli company M-Systems.
Israel has the highest number of personal computers per
capita in the world.
Israel has the highest number of university degrees per
capita in the world.
Israel produces more scientific papers per capita than any
other nation by a large margin - 109 per 10,000 people - as
well as one of the highest per capita rates of patents
filed.
In proportion to its population, Israel has the largest
number of startup companies in the world. In absolute terms,
Israel has the largest number of startup companies than any
other country in the world, except the US.
With more than 3,000 high-tech companies and startups,
Israel has the highest concentration of hi-tech companies in
the world - apart from Silicon Valley.
Israel is ranked #2 in the world for venture capital funds
right behind the United States.
Outside the United States and Canada, Israel has the largest
number of companies listed on NASDAQ.
Israel has the highest average living standards in the
Middle East. The per capita income in 2000 was over $17,500,
exceeding that of the United Kingdom.
On a per capita basis, Israel has the largest number of
biotech startups.
Twenty four percent of Israel's workforce holds university
degrees - ranking third in the industrialized world, after
the United States and Holland - and 12 percent hold advanced
degrees.
Israel has the third highest rate of entrepreneurship - and
the highest rate among women and among people over 55 in the
world.
Relative to its population, Israel is the largest
immigrant-absorbing nation on earth.
Israel has the world's second highest supply of new books
per capita.
Israel has more museums per capita than any other country.
Israeli scientists developed the first fully computerized,
no-radiation diagnostic instrumentation for breast cancer.
An Israeli company developed a computerized system for
ensuring proper administration of medications, thus removing
human error from medical treatment. Every year in U. S.
hospitals 7,000 patients die from treatment mistakes.
Israel's Given Imaging developed the PillCam - the first
ingestible video camera, which is so small it fits inside a
pill. Used to view the small intestine from the inside, the
camera helps doctors diagnose cancer and digestive
disorders.
Researchers in Israel developed a new device that directly
helps the heart pump blood. The new device is synchronized
with the heart's mechanical operations through a
sophisticated system of sensors.
Israel leads the world in the number of scientists and
technicians in the workforce, with 145 per 10,000, as
opposed to 85 in the U.S., over 70 in Japan, and less than
60 in Germany.
A new acne treatment developed in Israel causes acne
bacteria to self-destruct - all without damaging
surroundings skin or tissue.
An Israeli company was the first to develop and install a
large-scale solar-powered and fully functional electricity
generating plant in Southern California's Mojave Desert.
The first computer anti-virus software package was developed
in Israel back in the 1970's.
Major law enforcement agencies use Israeli technologies to
monitor voices and messages on conventional phones, mobile
phones and e-mails.
An Israeli company, Teva, is the world's largest generic
pharmaceutical company.
A new brain implant has been developed in Israel that can
lower the risk of stroke by diverting blood clots away from
sensitive areas of the brain.
IBM scientists in Israel are playing a vital role in a
massive project of the European Organization for Nuclear
Research (CERN) to discover the origins of life on earth.
Israeli software company Check Point is the global leader in
Virtual Private Network (VPN) and firewall technologies.
Israeli company Elta is responsible for the world's first
civilian aircraft equipped with technology designed to
protect airliners from a missile attack.
Mashav, the Israeli Foreign Ministry's Center for
International Cooperation has trained over 200,000
international aid workers that have traveled to dozens of
countries to help with medicine, agriculture, disaster
relief, and many other issues.
Israel has, for many years, held the world record in milk
production.
Rummikub, the third highest selling board game in the world,
is manufactured in a family-run plant in the small southern
Israeli town of Arad.
Drip irrigation - the system that is based on using plastic
pipes that release small amounts of water next to crops or
plants - was developed by the Israeli engineer Simcha Blas
in the 1970's. The invention caused a revolution in
agriculture.
A design submitted by Israeli-born Michael Arad has been
chosen for the World Trade Center Memorial, from amongst
5,000 entries from around the world.
Israeli company Retalix created the grocery scanners used at
such stores as Costco, Albertson's, and 7-11, as well as
25,000 additional stores and quick-service restaurants
throughout the United States.
Primate research at Hebrew University is leading to the
development of a robotic arm that can respond to the brain
commands of a paralyzed person.
Two Israeli researchers are generating cancer-killing
molecules that will recognize cancerous cells and target
them aggressively, while not affecting normal cells.
Israeli researchers developed a novel stem cell therapy to
treat Parkinson's Disease - using a patient's own bone
marrow stem cells to produce the missing chemical that
enables restoration of motor movement.
Israeli company Silent Communications has developed a type
of silent conversation system for cell phones, so users can
carry on conversations without saying a word.
The Israeli company Wondernet is currently dominating the
world market in document signature authentication, with its
unique scientific method of verifying handwritten
signatures.
Israeli Professor Yehuda Finkelstein has discovered the
cause of and cure for halitosis (bad breath).
Cherry tomatoes were originally supposed to be a snack when
they were designed by a group of scientists led by professor
Nahum Keidar from the agriculture faculty at the Weizmann
Institute of Science, with the cooperation of the Israeli
company Zera.
The Quicktionary, a pen size scanner that scans a word or a
sentence and translates it to a different language, was
developed by the Wizcom Company, based in Jerusalem.
Professor Ehud Keinan from the Technion Israel Institute of
Technology developed a pen that identifies an improvised
explosive.
The Israeli company Insightec developed an ultrasound system
for removing tumors without surgery.
Researchers at the Technion have developed an antibiotic
that destroys anthrax bacteria as well as the toxins it
secretes into the bloodstream of the infected body.
Epilady, an electric hair removal system, was developed by
Yair Dar and Shimon Yahav from the Goshrim Kibbutz.
The sun-heated water tank, a device that converts solar
energy into thermal energy and that saves about 4% of the
national energy supply, was developed by an engineer from
Jerusalem.
Dr. Gal Yadid, Dr. Rachel Mayan, and Professor Abraham
Weizman from Bar Ilan University developed a form of drug
rehabilitation using a natural steroid that is inserted into
the brain and develops a resistance for the drugs.
Alon Moses from Hadassah Medical Center in Jerusalem and
Imanuel Hensky and Carlos Hidelgo-Grass from Hebrew
University decoded the mechanism for Streptococcus A.
Wednesday, July 7, 2010
Number Theory
Number Theory
Elementary number theory provides a rich set of tools for the implementation of cryptographic schemes. Most publickey cryptosystems are based in one way or another on numbertheoretic id
Bignum computations
Many cryptographic schemes, such as RSA, work with large integers, also known as “bignums” or “multi
precision integers.” Here “large” may mean 160–4096 bits (49–1233 decimal digits), with 1024bit integers
(308 decimal digits) typical. We briefly overview of some implementation issues and possibilities.
When RSA was invented, efficiently implementing it was a problem. Today, standard desktop CPU’s perform bignum computations quickly. Still, for servers doing hundreds of SSL connections per second, a hardware assist may be needed, such as the SSL accelerators produced by nCipher www.ncipher.com/.
A popular C/C++ software subroutine library supporting multiprecision operations is GMP (GNU Multi
precision package) www.swox.com/gmp/. A more elaborate package (based on GMP) is Shoup’s NTL
(Number Theory Library) www.shoup.net/ntl/. For a survey, see
https://www.cosic.esat.kuleuven.ac.be/nessie/call/mplibs.html.
Java has excellent support for multiprecision operations in its BigInteger class java.sun.com/j2se/1.4.1/docs/api/java/math/BigInteger.html; this includes a primality testing routine.
Python www.python.org/ is a personal favorite; it includes direct support for large integers.
Scheme www.swiss.ai.mit.edu/projects/scheme/ also provides direct bignum support.
Some other pointers to software and hardware implementations can be found in the “Practical Aspects”
section of Helger Lipmaa’s “Cryptology pointers” www.tcs.hut.fi/˜helger/crypto/=.
When working on kbit integers, most implementations implement addition and subtraction in time O(k), multiplication, division, and gcd in time O(k2 ) (although faster implementations exist for very large k), and modular exponentation in time O(k3 ).
To get you roughly calibrated, here are some timings, obtained from a simple Python program on my IBM Thinkpad laptop (1.2 GHz PIII processor) on 1024bit inputs. SHA1 is included just for comparison. The last column gives the approximate ratio of running time to addition.
2.2 microseconds addition 455,000 per second 1
4.4 microseconds SHA1 hash (on 20byte input) 227,000 per second 2
10.8 microseconds modular addition 93,000 per second 5
41 microseconds multiplication 24,000 per second 20
135 microseconds modular multiplication 7,400 per second 60
2.3 milliseconds modular exponentiation (exponent is 2**16+1) 440 per second 1000
5.5 milliseconds gcd 180 per second 2500
204 milliseconds modular exponentiation (1024bit exponent) 5 per second 93000
Divisors and Divisibility
Definition 1 (Divides relation, divisor, common divisor) We say that “d divides a”, written d | a, if there exists an
integer k such that a = kd. If d does not divide a, we write “d
If d | a and d | b, then d is a common divisor of a and b.
| a”. If d | a and d ≥ 0, we say that d is a divisor of a.
Example 1 Every integer d ≥ 0 (including d = 0) is a divisor of 0. While 0 divides no integer except itself, 1 is a divisor of every integer. The divisors of 12 are {1, 2, 3, 4, 6, 12}. A common divisor of 14 and 77 is 7. If d | a then d | (−a).
Definition 2 (prime) An integer p > 1 is prime if its only divisors are 1 and p.
Definition 3 (Greatest common divisor, relatively prime) The greatest common divisor, gcd(a, b), of two integers a and b is the largest of their common divisors, except that gcd(0, 0) = 0by definition. Integers a and b are relatively prime if gcd(a, b) =. 1
Example 2
gcd(24, 30) = 6 gcd(4, 7) = 1 gcd(0, 5) = 5 gcd(−6, 10) = 2
Example 3 For all a ≥ 0, a and a + 1are relatively prime. The integer 1 is relatively prime to all other integers.
Example 4 If p is prime and 1 ≤ a < p, then gcd(a, p) =. 1That is, a and p are relatively prime.
Definition 4 For any positive integer n, we define Euler’s phi function of n, denoted φ(n), as the number of integers d,
1 ≤ d ≤ n, that are relatively prime to n. (Note that φ(1) = 1.)
Example 5 If p is prime, then φ(p) = p − 1. For any integer k > 0, φ(2k ) =k 2−1 .
Definition 5 The least common multiple lcm(a, b) of two integers a ≥ 0, b ≥ 0, is the least m such that a | m and
b | m.
Exercise 1 Show that the number of divisors of n = pe1 e2 ek �
ei ).
Exercise 2 Show that lcm(a, b) = ab/ gcd(a, b).
1 p2
• • • pk
(where the pi ’s are distinct primes) is
1≤i≤k (1 +
Fermat’s Little Theorem
Theorem 1 (Fermat’s Little Theorem) If p is prime and a ∈ Z∗
p , then ap−1 = 1 (mop) d.
Theorem 2 (Lagrange’s Theorem) The order of a subgroup must divide the order of a group.
Fermat’s Little Theorem follows from Lagrange’s Theorem, since the order of the subgroup a generated by a in Z∗ is
the least t > 0 such that at = 1 (mod p), and Z∗ = p − 1.
�� p
| p |
Euler’s Theorem generalizes Fermat’s Little Theorem, since Z∗ = φ(n) for all n > 0.
| n |
Theorem 3 (Euler’s Theorem) For any n > 1 and any a ∈ Z∗
n , aφ(n) = 1 (modn) .
A somewhat tighter result actually holds. Define for n > 0 Carmichael’s lambda function λ(n) to be the least positive
t such that at
= 1 (mod n) for all a ∈ Z∗
n . Then λ(1) =λ(2) = , 1λ(4) = , 2λ(2e ) = e 2−2 for e > 2,
λ(pe ) = pe−1 (p − 1) if p is an odd prime, and if n = pe1 • • • pek , then
1 k
λ(n) =
λ(pe1lcm(
ek )) .
1 ), . . . , λ(pk
Computing modular inverses. Fermat’s Little Theorem provides a convenient way to compute the modular inverse a−1
(mod p) for any a ∈ Z∗
p , where p is prime:
a−1 = ap−2
(mod p) .
(Euclid’s extended algorithm for computing gcd(a, p) is more efficient.)
Primality testing. The converse of Fermat’s Little Theorem is “almost” true. The converse would say that if 1 ≤ a < p
and ap−1 = 1 (mod p), then p is prime. Suppose that p is a large randomly chosen integer, and that a is a randomly chosen integer such that 1 ≤ a < p. Then if ap−1 = 1 (mod p), then p is certainly not prime (by FLT), and otherwise p is “likely” to be prime. FLT thus provides a heuristic test for primality for randomly chosen p; refinements of this approach yield tests effective for all p.
Exercise 1 Prove that λ(n) is always a divisor of φ(n), and characterize exactly when it is a proper divisor.
Exercise 2 Suppose a > 1 is not even or divisible by 5; show that a100 (in decimal) ends in 001.
Exercise 3 Let p be prime. (a) Show that ap = a (mod p) for any a ∈ Zp . (b) Argue that (a + b)p = ap + bp
for any a,b in Zp . (c) Show that (me )d = m (mod p) for all m ∈ Zp if ed = 1 (mod p − 1).
(mod p)
Generators
Definition 1 A finite group G = (S, •) may be cyclic, which means that it contains a generator g such that every group element h ∈ S is a power h = gk of g for some k ≥ 0. If the group operation is addition, we write this condition as
g + g + • • • + g = kg .h =
� �� �
k
Example 1 For example, 3 generates Z10 under addition, since the multiples of 3, modulo 10, are:
3, 6, 9, 2, 5, 8, 1, 4, 7, 0 .
Fact 1 The generators of (Zm , +) are exactly those φ(m) integers a ∈ Zm relatively prime to m.
Example 2 The generators of (Z10 , +) are {1, 3, 7, 9}.
Example 3 The group (Z∗
11 , •) is generated by g = 2, since the powers of 2 (modulo 11) are:
2, 4, 8, 5, 10, 9, 7, 3, 6, 1 .
Fact 2 Any cyclic group of size m is isomorphic to (Zm , +). For example, (Z∗ ) ↔ (Z10 , +) via:
11 , •
2x (mod 11) ←→ x (mod 10) .
Theorem 1 If p is prime, then (Z ∗ ) is cyclic, and contains φ(p − 1) generators. More generally, the group (Zn , •) is
p , •
cyclic if and only if n = 2, n = 4, n = pe , or n = 2pe , where p is an odd prime and e ≥ 1; in these cases the group contains φ(φ(n)) generators.
Finding a generator of Z∗
p . If the factorization of p − 1 is unknown, no efficient algorithm is known, but if p − 1 has known factorization, it is easy to find a generator. Generators of Z∗
p are relatively common (φ(n) ≥ n/(6 ln n ln) for n ≥ 5), so one can be found by searching at random for an element g whose order is p − 1. (Note g has order p − 1 if gp−1 = 1 (mod p) but g(p−1)/q = 1 (mod p) for all prime divisors q of p − 1).
Group generated by an element. In any group G, the set �g� of elements generated by g is always a cyclic subgroup of
G; if �g� = G then g is a generator of G.
Groups of prime order. If a group H has prime order, then every element except the identity is a generator. For example, the subgroup QR11 = {1, 4, 9, 5, 3} of squares (quadratic residues) in Z∗
11 has order 5, so 4, 9, 5, and 3 all generate QR11 . For this reason, it is sometimes of interest to work with the group QRp of squares modulo p, where p = 2q + 1 and q is prime.
∗
p , •); prove that gxExercise 1 (a) Find all of the generators of (Z11 , •) and of (Z2k , +). (b) Let g be a generator of (Z
generates Z∗
p if and only if x generates (Zp−1 , +).
Orders of Elements
Definition 1 The order of an element a of a finite group G is the least positive t such that at = 1. (If the group is written additively, it is the least positive t such that a + a + • • • + a (t times) = 0.)
1 2 3 4 5 6 7 . . . Order
1 1 1 1 1 1 1 1 . . . 1
Row a column k contains ak mod p for p = 7; bold
k
2 2 4 1 2 4 1 2 . . . 3
3 3 2 6 4 5 1 3 . . . 6
4 4 2 1 4 2 1 4 . . . 3
5 5 4 6 2 3 1 5 . . . 6
face entries illustrate the fundamental period of a
(mod p) as k increases. The length of this period is the order of a, modulo p. By Fermat’s Little The orem the order always divides p − 1; thus ap−1 is always 1 (see the column marked with an uparrow). Elements 3 and 5 have order p − 1, and so are gen erators of Z∗
7 . Element 6 is −1, modulo 7, and thus
6 6 1 6 1 6 1 6 . . . 2
⇑
has order 2.
Fact 1 The order of an element a ∈ G is a divisor of the order of G. (The order |G| of a group G is the number of
elements it contains.) Therefore a|G| = 1 in G . Thus when p is prime, the order of an element a ∈ Z∗ is a divisor of
p
Z∗ = p − 1, and in general the order of an element a ∈ Z∗ is a divisor of Z∗ = φ(n).
| p |
|n n |
Computing the order t of an element a ∈ G. If the factorization of |G| is unknown, no efficient algorithm is known,
but if G has known factorization G = pe1 e2 ek
1f f2 fk
| | | |
1 p2
• • • pk , it is easy. Basically, compute the order t as t = p1 p2
• • • pk
where each fi is initially ei , then each fi is decreased in turn as much as possible (but not below zero) while keeping
at = 1 in G.
Fact 2 When p is prime, the number of elements in Z∗ of order d, where d | (p − 1), is φ(d). For example, since
p
φ(2) =, 1there is a unique square root of 1 modulo p, other than 1 itself (it is −1 = p − 1 (modp)).
Exercise 1 Let ord(a) denote the order of a ∈ G. (a) Prove that ord(a) = ord(a−1 ) and ord(ak ) |
ord(a). (b) Prove
that ord(ab) is a divisor of lcm(ord(a), ord(b)), and show that it may be a proper divisor. (c) Show that ord(ab) =
ord(a) ord(b) if gcd(ord(a), ord(b)) =. 1
Exercise 2 Show that there are at least as many elements of order p − 1 (i.e. generators) of Z∗ as there are elements of
p
any other order.
Exercise 3 Show that the order of a in (Zn , +) is n/ gcd(a, n).
Euclid’s Algorithm for Computing GCD
It is easy to compute gcd(a, b). This is surprising because you might think that in order to compute gcd(a, b) you would need to figure out their divisors, i.e. solve the factoring problem. But, as you will see, we don’t need to figure out the divisors of a and b to find their gcd.
Euclid (circa 300 B.C.) showed how to compute gcd(a, b) for a ≥ 0 and b ≥ 0:
�
gcd(a, b) =
a if b = 0
gcd(b, a mod b) otherwise
The recursion terminates since (a mod b) < b; the second argument strictly decreases with each call. An equivalent nonrecursive version sets a0 = a, a1 = b, and then computes ai+1 for i = 2, 3, . . . as ai+1 = ai−1 mod ai until ai+1 = 0, then returns ai .
Example 1 Euclid’s Algorithm finds the greatest common divisor of 12 and 33 as:
gcd(12, 33) = gcd(33, 12) = gcd(12, 9) = gcd(9, 3) = gcd(3, 0) = 3.
The equivalent nonrecursive version has a0 = 12, a1 = 33, and
a2 = a0 mod a1 = 12 mod 33 = 12 a3 = a1 mod a2 = 33 mod 12 = 9 a4 = a2 mod a3 = 12 mod 9 = 3
a5 = a3 mod a4 = 9 mod 3 = 0
So gcd(12, 33) =. 3
It can be shown that the number of recursive calls is O(log b); the worstcase input is a pair of consecutive Fibonacci numbers. Euclid’s algorithm (even if extended) takes O(k2 ) bit operations when inputs a and b have at most k bits; see Bach and Shallit.
Euclid’s Extended Algorithm
Theorem 1 For all integers a, b, one can efficiently compute integers x and y such that
gcd(a, b) = ax + by .
We give a “proof by example,” using Euclid’s Extended Algorithm on inputs a = 9, b = 31, which for each ai of the nonrecursive version of Euclid’s algorithm finds an xi and yi such that ai = axi + byi :
a0 = a = 9 = a ∗ 1 +b ∗ 0
a1 = b = 31 = a ∗ 0 +b ∗ 1
2a =
3a =
4a =
5a =
a0 mod a1 =
a1 mod a2 =
a2 mod a3 =
a3 mod a4 =
=9 (a ∗ 1 +b ∗ 0) − 0 ∗ (a ∗ 0 +b ∗ 1) =
=4 (a ∗ 0 +b ∗ 1) − 3 ∗ (a ∗ 1 +b ∗ 0) =
=1 (a ∗ 1 +b ∗ 0) − 2 ∗ (a ∗ (−3) +b ∗ 1) =
0
a ∗ 1 +b ∗ 0 a ∗ (−3) +b ∗ 1 a ∗ 7 +b ∗ (−2)
Thus Euclid’s Extended Algorithm computes x = 7 and y = −2 for a = 9 and b = 31.
Corollary 1 (Multiplicative inverse computation) Given integers n and a where gcd(a, n) =, 1using Euclid’s Ex tended Algorithm to find x and y such that ax + ny = 1 finds an x such that ax ≡ 1 (modn); such an x is the multiplicative inverse of a modulo n: x = a−1 (mod n).
Example 1 The multiplicative inverse of 9, modulo 31, is 7. Check: 9 ∗ 7 = 63 = 1mod (31).
Exercise 1 Find the multiplicative inverse of 11 modulo 41.
Exercise 2 Prove that if gcd(a, n) > 1, then the multiplicative inverse a−1 (mod n) does not exist.
Exercise 3 Show that Euclid’s algorithm is correct by arguing that d is a common divisor of a and b if and only if d is a common divisor of b and (a mod b).
Chinese Remainder Theorem
When working modulo a composite modulus n, the Chinese Remainder Theorem (CRT) can both speed computation modulo n and facilitate reasoning about the properties of arithmetic modulo n.
Theorem 1 (Chinese Remainder Theorem (CRT)) Let n = n1 n2 • • • nk be the product of k integers ni that are pair
wise relatively prime. The mapping
f (a) =a(1 , . . . , ak ) =a(mod n1 , . . . , a mod nk )
is an isomorphism from Zn to Zn1 × • • • × Znk : if f (a) =a(1 , . . . , ak ) and f (b) =b (1 , . . . , bk ), then
f ((a ± b) mod n) = ((a1 ± b1 ) mod n1 , . . . , (ak ± bk ) mod nk )
f ((ab) mod n) = ((a1 b1 ) mod n1 , . . . , (ak bk ) mod nk )
f (a−1 mod n) = a(−1 mod n1 , . . . , a−1 mod nk ) if a−1 (mod n) exists
1
f −1 ((a1 , . . . , ak )) = a =
� k
ai ci (mod n) where mi = n/ni and ci = mi (m−1 mod ni ) .
i i
When n = pq is the product of two primes, working modulo n is equivalent to working independently on each component of its CRT (i.e. (mod p, mod q)) representation. It can be worthwhile to convert an input to its CRT representation, compute in that representation, and then convert back.
Example: For n = 35 = 5•
7put (a mod 35) in row a1 = (a mod 5) and column a2 = (a mod 7):
0 1 2 3 4 5 6
f (8) = (3, 1)
0 0 15 30 10 25 5 20
1 21 1 16 31 11 26 6
f (−8) =f (27)
f (12)
= (−3, −1) = , (26)
= (2, 5)
2 7 22 2 17 32 12 27
3 28 8 23 3 18 33 13
4 14 29 9 24 4 19 34
f (12−1 ) = (2−1 , 5−1 ) = , (33) =f (3)
f (8 + 12)f =(20) = (3 +, 1 2 + 5) , =6) (0
f (8 • 12) =f (96) =f (26) = (3 • 2, 1 • 5) = , (15)
Here m1 = 7, m2 = 5, c1 = 7 • (7−1 mod 5) = 7•
3 =, c212 = 5 • (5−1 mod 7) = 5•
3 =, 15so
f −1 ((a1 , a2 )) = 21a1 + 15a2 (mod 35).
(Note: f (21) = , (10), f (15) = , (01).) Thus, f −1 ((1, 5)) = 21 +•
5 15 = 96mod =35). 26 (
Speeding up Modular Exponentation. A significant application is speeding up exponentiation modulo n = pq when p
and q are known. To compute y = xd mod n, where f (x) =x(1 , x2 ):
f (y) = f (xd ) =xd( d
d mod (p−1) mod p, xd mod (q−1) mod q) .
Note xp−1
1 mod p, x2 mod q) =x(1 2
1 = 1 mod p for x1 = 0 by Fermat’s Little Theorem. Then convert back from (y mod p, y mod q) to y mod n. Since exponentiation takes time cubic in the input size, two halfsize exponentiations are about four times faster than one fullsize exponentiation (including conversion).
Exercise 1 Prove that x is a square mod n = pq if and only if it is a square mod p and mod q.
Elementary number theory provides a rich set of tools for the implementation of cryptographic schemes. Most publickey cryptosystems are based in one way or another on numbertheoretic id
Bignum computations
Many cryptographic schemes, such as RSA, work with large integers, also known as “bignums” or “multi
precision integers.” Here “large” may mean 160–4096 bits (49–1233 decimal digits), with 1024bit integers
(308 decimal digits) typical. We briefly overview of some implementation issues and possibilities.
When RSA was invented, efficiently implementing it was a problem. Today, standard desktop CPU’s perform bignum computations quickly. Still, for servers doing hundreds of SSL connections per second, a hardware assist may be needed, such as the SSL accelerators produced by nCipher www.ncipher.com/.
A popular C/C++ software subroutine library supporting multiprecision operations is GMP (GNU Multi
precision package) www.swox.com/gmp/. A more elaborate package (based on GMP) is Shoup’s NTL
(Number Theory Library) www.shoup.net/ntl/. For a survey, see
https://www.cosic.esat.kuleuven.ac.be/nessie/call/mplibs.html.
Java has excellent support for multiprecision operations in its BigInteger class java.sun.com/j2se/1.4.1/docs/api/java/math/BigInteger.html; this includes a primality testing routine.
Python www.python.org/ is a personal favorite; it includes direct support for large integers.
Scheme www.swiss.ai.mit.edu/projects/scheme/ also provides direct bignum support.
Some other pointers to software and hardware implementations can be found in the “Practical Aspects”
section of Helger Lipmaa’s “Cryptology pointers” www.tcs.hut.fi/˜helger/crypto/=.
When working on kbit integers, most implementations implement addition and subtraction in time O(k), multiplication, division, and gcd in time O(k2 ) (although faster implementations exist for very large k), and modular exponentation in time O(k3 ).
To get you roughly calibrated, here are some timings, obtained from a simple Python program on my IBM Thinkpad laptop (1.2 GHz PIII processor) on 1024bit inputs. SHA1 is included just for comparison. The last column gives the approximate ratio of running time to addition.
2.2 microseconds addition 455,000 per second 1
4.4 microseconds SHA1 hash (on 20byte input) 227,000 per second 2
10.8 microseconds modular addition 93,000 per second 5
41 microseconds multiplication 24,000 per second 20
135 microseconds modular multiplication 7,400 per second 60
2.3 milliseconds modular exponentiation (exponent is 2**16+1) 440 per second 1000
5.5 milliseconds gcd 180 per second 2500
204 milliseconds modular exponentiation (1024bit exponent) 5 per second 93000
Divisors and Divisibility
Definition 1 (Divides relation, divisor, common divisor) We say that “d divides a”, written d | a, if there exists an
integer k such that a = kd. If d does not divide a, we write “d
If d | a and d | b, then d is a common divisor of a and b.
| a”. If d | a and d ≥ 0, we say that d is a divisor of a.
Example 1 Every integer d ≥ 0 (including d = 0) is a divisor of 0. While 0 divides no integer except itself, 1 is a divisor of every integer. The divisors of 12 are {1, 2, 3, 4, 6, 12}. A common divisor of 14 and 77 is 7. If d | a then d | (−a).
Definition 2 (prime) An integer p > 1 is prime if its only divisors are 1 and p.
Definition 3 (Greatest common divisor, relatively prime) The greatest common divisor, gcd(a, b), of two integers a and b is the largest of their common divisors, except that gcd(0, 0) = 0by definition. Integers a and b are relatively prime if gcd(a, b) =. 1
Example 2
gcd(24, 30) = 6 gcd(4, 7) = 1 gcd(0, 5) = 5 gcd(−6, 10) = 2
Example 3 For all a ≥ 0, a and a + 1are relatively prime. The integer 1 is relatively prime to all other integers.
Example 4 If p is prime and 1 ≤ a < p, then gcd(a, p) =. 1That is, a and p are relatively prime.
Definition 4 For any positive integer n, we define Euler’s phi function of n, denoted φ(n), as the number of integers d,
1 ≤ d ≤ n, that are relatively prime to n. (Note that φ(1) = 1.)
Example 5 If p is prime, then φ(p) = p − 1. For any integer k > 0, φ(2k ) =k 2−1 .
Definition 5 The least common multiple lcm(a, b) of two integers a ≥ 0, b ≥ 0, is the least m such that a | m and
b | m.
Exercise 1 Show that the number of divisors of n = pe1 e2 ek �
ei ).
Exercise 2 Show that lcm(a, b) = ab/ gcd(a, b).
1 p2
• • • pk
(where the pi ’s are distinct primes) is
1≤i≤k (1 +
Fermat’s Little Theorem
Theorem 1 (Fermat’s Little Theorem) If p is prime and a ∈ Z∗
p , then ap−1 = 1 (mop) d.
Theorem 2 (Lagrange’s Theorem) The order of a subgroup must divide the order of a group.
Fermat’s Little Theorem follows from Lagrange’s Theorem, since the order of the subgroup a generated by a in Z∗ is
the least t > 0 such that at = 1 (mod p), and Z∗ = p − 1.
�� p
| p |
Euler’s Theorem generalizes Fermat’s Little Theorem, since Z∗ = φ(n) for all n > 0.
| n |
Theorem 3 (Euler’s Theorem) For any n > 1 and any a ∈ Z∗
n , aφ(n) = 1 (modn) .
A somewhat tighter result actually holds. Define for n > 0 Carmichael’s lambda function λ(n) to be the least positive
t such that at
= 1 (mod n) for all a ∈ Z∗
n . Then λ(1) =λ(2) = , 1λ(4) = , 2λ(2e ) = e 2−2 for e > 2,
λ(pe ) = pe−1 (p − 1) if p is an odd prime, and if n = pe1 • • • pek , then
1 k
λ(n) =
λ(pe1lcm(
ek )) .
1 ), . . . , λ(pk
Computing modular inverses. Fermat’s Little Theorem provides a convenient way to compute the modular inverse a−1
(mod p) for any a ∈ Z∗
p , where p is prime:
a−1 = ap−2
(mod p) .
(Euclid’s extended algorithm for computing gcd(a, p) is more efficient.)
Primality testing. The converse of Fermat’s Little Theorem is “almost” true. The converse would say that if 1 ≤ a < p
and ap−1 = 1 (mod p), then p is prime. Suppose that p is a large randomly chosen integer, and that a is a randomly chosen integer such that 1 ≤ a < p. Then if ap−1 = 1 (mod p), then p is certainly not prime (by FLT), and otherwise p is “likely” to be prime. FLT thus provides a heuristic test for primality for randomly chosen p; refinements of this approach yield tests effective for all p.
Exercise 1 Prove that λ(n) is always a divisor of φ(n), and characterize exactly when it is a proper divisor.
Exercise 2 Suppose a > 1 is not even or divisible by 5; show that a100 (in decimal) ends in 001.
Exercise 3 Let p be prime. (a) Show that ap = a (mod p) for any a ∈ Zp . (b) Argue that (a + b)p = ap + bp
for any a,b in Zp . (c) Show that (me )d = m (mod p) for all m ∈ Zp if ed = 1 (mod p − 1).
(mod p)
Generators
Definition 1 A finite group G = (S, •) may be cyclic, which means that it contains a generator g such that every group element h ∈ S is a power h = gk of g for some k ≥ 0. If the group operation is addition, we write this condition as
g + g + • • • + g = kg .h =
� �� �
k
Example 1 For example, 3 generates Z10 under addition, since the multiples of 3, modulo 10, are:
3, 6, 9, 2, 5, 8, 1, 4, 7, 0 .
Fact 1 The generators of (Zm , +) are exactly those φ(m) integers a ∈ Zm relatively prime to m.
Example 2 The generators of (Z10 , +) are {1, 3, 7, 9}.
Example 3 The group (Z∗
11 , •) is generated by g = 2, since the powers of 2 (modulo 11) are:
2, 4, 8, 5, 10, 9, 7, 3, 6, 1 .
Fact 2 Any cyclic group of size m is isomorphic to (Zm , +). For example, (Z∗ ) ↔ (Z10 , +) via:
11 , •
2x (mod 11) ←→ x (mod 10) .
Theorem 1 If p is prime, then (Z ∗ ) is cyclic, and contains φ(p − 1) generators. More generally, the group (Zn , •) is
p , •
cyclic if and only if n = 2, n = 4, n = pe , or n = 2pe , where p is an odd prime and e ≥ 1; in these cases the group contains φ(φ(n)) generators.
Finding a generator of Z∗
p . If the factorization of p − 1 is unknown, no efficient algorithm is known, but if p − 1 has known factorization, it is easy to find a generator. Generators of Z∗
p are relatively common (φ(n) ≥ n/(6 ln n ln) for n ≥ 5), so one can be found by searching at random for an element g whose order is p − 1. (Note g has order p − 1 if gp−1 = 1 (mod p) but g(p−1)/q = 1 (mod p) for all prime divisors q of p − 1).
Group generated by an element. In any group G, the set �g� of elements generated by g is always a cyclic subgroup of
G; if �g� = G then g is a generator of G.
Groups of prime order. If a group H has prime order, then every element except the identity is a generator. For example, the subgroup QR11 = {1, 4, 9, 5, 3} of squares (quadratic residues) in Z∗
11 has order 5, so 4, 9, 5, and 3 all generate QR11 . For this reason, it is sometimes of interest to work with the group QRp of squares modulo p, where p = 2q + 1 and q is prime.
∗
p , •); prove that gxExercise 1 (a) Find all of the generators of (Z11 , •) and of (Z2k , +). (b) Let g be a generator of (Z
generates Z∗
p if and only if x generates (Zp−1 , +).
Orders of Elements
Definition 1 The order of an element a of a finite group G is the least positive t such that at = 1. (If the group is written additively, it is the least positive t such that a + a + • • • + a (t times) = 0.)
1 2 3 4 5 6 7 . . . Order
1 1 1 1 1 1 1 1 . . . 1
Row a column k contains ak mod p for p = 7; bold
k
2 2 4 1 2 4 1 2 . . . 3
3 3 2 6 4 5 1 3 . . . 6
4 4 2 1 4 2 1 4 . . . 3
5 5 4 6 2 3 1 5 . . . 6
face entries illustrate the fundamental period of a
(mod p) as k increases. The length of this period is the order of a, modulo p. By Fermat’s Little The orem the order always divides p − 1; thus ap−1 is always 1 (see the column marked with an uparrow). Elements 3 and 5 have order p − 1, and so are gen erators of Z∗
7 . Element 6 is −1, modulo 7, and thus
6 6 1 6 1 6 1 6 . . . 2
⇑
has order 2.
Fact 1 The order of an element a ∈ G is a divisor of the order of G. (The order |G| of a group G is the number of
elements it contains.) Therefore a|G| = 1 in G . Thus when p is prime, the order of an element a ∈ Z∗ is a divisor of
p
Z∗ = p − 1, and in general the order of an element a ∈ Z∗ is a divisor of Z∗ = φ(n).
| p |
|n n |
Computing the order t of an element a ∈ G. If the factorization of |G| is unknown, no efficient algorithm is known,
but if G has known factorization G = pe1 e2 ek
1f f2 fk
| | | |
1 p2
• • • pk , it is easy. Basically, compute the order t as t = p1 p2
• • • pk
where each fi is initially ei , then each fi is decreased in turn as much as possible (but not below zero) while keeping
at = 1 in G.
Fact 2 When p is prime, the number of elements in Z∗ of order d, where d | (p − 1), is φ(d). For example, since
p
φ(2) =, 1there is a unique square root of 1 modulo p, other than 1 itself (it is −1 = p − 1 (modp)).
Exercise 1 Let ord(a) denote the order of a ∈ G. (a) Prove that ord(a) = ord(a−1 ) and ord(ak ) |
ord(a). (b) Prove
that ord(ab) is a divisor of lcm(ord(a), ord(b)), and show that it may be a proper divisor. (c) Show that ord(ab) =
ord(a) ord(b) if gcd(ord(a), ord(b)) =. 1
Exercise 2 Show that there are at least as many elements of order p − 1 (i.e. generators) of Z∗ as there are elements of
p
any other order.
Exercise 3 Show that the order of a in (Zn , +) is n/ gcd(a, n).
Euclid’s Algorithm for Computing GCD
It is easy to compute gcd(a, b). This is surprising because you might think that in order to compute gcd(a, b) you would need to figure out their divisors, i.e. solve the factoring problem. But, as you will see, we don’t need to figure out the divisors of a and b to find their gcd.
Euclid (circa 300 B.C.) showed how to compute gcd(a, b) for a ≥ 0 and b ≥ 0:
�
gcd(a, b) =
a if b = 0
gcd(b, a mod b) otherwise
The recursion terminates since (a mod b) < b; the second argument strictly decreases with each call. An equivalent nonrecursive version sets a0 = a, a1 = b, and then computes ai+1 for i = 2, 3, . . . as ai+1 = ai−1 mod ai until ai+1 = 0, then returns ai .
Example 1 Euclid’s Algorithm finds the greatest common divisor of 12 and 33 as:
gcd(12, 33) = gcd(33, 12) = gcd(12, 9) = gcd(9, 3) = gcd(3, 0) = 3.
The equivalent nonrecursive version has a0 = 12, a1 = 33, and
a2 = a0 mod a1 = 12 mod 33 = 12 a3 = a1 mod a2 = 33 mod 12 = 9 a4 = a2 mod a3 = 12 mod 9 = 3
a5 = a3 mod a4 = 9 mod 3 = 0
So gcd(12, 33) =. 3
It can be shown that the number of recursive calls is O(log b); the worstcase input is a pair of consecutive Fibonacci numbers. Euclid’s algorithm (even if extended) takes O(k2 ) bit operations when inputs a and b have at most k bits; see Bach and Shallit.
Euclid’s Extended Algorithm
Theorem 1 For all integers a, b, one can efficiently compute integers x and y such that
gcd(a, b) = ax + by .
We give a “proof by example,” using Euclid’s Extended Algorithm on inputs a = 9, b = 31, which for each ai of the nonrecursive version of Euclid’s algorithm finds an xi and yi such that ai = axi + byi :
a0 = a = 9 = a ∗ 1 +b ∗ 0
a1 = b = 31 = a ∗ 0 +b ∗ 1
2a =
3a =
4a =
5a =
a0 mod a1 =
a1 mod a2 =
a2 mod a3 =
a3 mod a4 =
=9 (a ∗ 1 +b ∗ 0) − 0 ∗ (a ∗ 0 +b ∗ 1) =
=4 (a ∗ 0 +b ∗ 1) − 3 ∗ (a ∗ 1 +b ∗ 0) =
=1 (a ∗ 1 +b ∗ 0) − 2 ∗ (a ∗ (−3) +b ∗ 1) =
0
a ∗ 1 +b ∗ 0 a ∗ (−3) +b ∗ 1 a ∗ 7 +b ∗ (−2)
Thus Euclid’s Extended Algorithm computes x = 7 and y = −2 for a = 9 and b = 31.
Corollary 1 (Multiplicative inverse computation) Given integers n and a where gcd(a, n) =, 1using Euclid’s Ex tended Algorithm to find x and y such that ax + ny = 1 finds an x such that ax ≡ 1 (modn); such an x is the multiplicative inverse of a modulo n: x = a−1 (mod n).
Example 1 The multiplicative inverse of 9, modulo 31, is 7. Check: 9 ∗ 7 = 63 = 1mod (31).
Exercise 1 Find the multiplicative inverse of 11 modulo 41.
Exercise 2 Prove that if gcd(a, n) > 1, then the multiplicative inverse a−1 (mod n) does not exist.
Exercise 3 Show that Euclid’s algorithm is correct by arguing that d is a common divisor of a and b if and only if d is a common divisor of b and (a mod b).
Chinese Remainder Theorem
When working modulo a composite modulus n, the Chinese Remainder Theorem (CRT) can both speed computation modulo n and facilitate reasoning about the properties of arithmetic modulo n.
Theorem 1 (Chinese Remainder Theorem (CRT)) Let n = n1 n2 • • • nk be the product of k integers ni that are pair
wise relatively prime. The mapping
f (a) =a(1 , . . . , ak ) =a(mod n1 , . . . , a mod nk )
is an isomorphism from Zn to Zn1 × • • • × Znk : if f (a) =a(1 , . . . , ak ) and f (b) =b (1 , . . . , bk ), then
f ((a ± b) mod n) = ((a1 ± b1 ) mod n1 , . . . , (ak ± bk ) mod nk )
f ((ab) mod n) = ((a1 b1 ) mod n1 , . . . , (ak bk ) mod nk )
f (a−1 mod n) = a(−1 mod n1 , . . . , a−1 mod nk ) if a−1 (mod n) exists
1
f −1 ((a1 , . . . , ak )) = a =
� k
ai ci (mod n) where mi = n/ni and ci = mi (m−1 mod ni ) .
i i
When n = pq is the product of two primes, working modulo n is equivalent to working independently on each component of its CRT (i.e. (mod p, mod q)) representation. It can be worthwhile to convert an input to its CRT representation, compute in that representation, and then convert back.
Example: For n = 35 = 5•
7put (a mod 35) in row a1 = (a mod 5) and column a2 = (a mod 7):
0 1 2 3 4 5 6
f (8) = (3, 1)
0 0 15 30 10 25 5 20
1 21 1 16 31 11 26 6
f (−8) =f (27)
f (12)
= (−3, −1) = , (26)
= (2, 5)
2 7 22 2 17 32 12 27
3 28 8 23 3 18 33 13
4 14 29 9 24 4 19 34
f (12−1 ) = (2−1 , 5−1 ) = , (33) =f (3)
f (8 + 12)f =(20) = (3 +, 1 2 + 5) , =6) (0
f (8 • 12) =f (96) =f (26) = (3 • 2, 1 • 5) = , (15)
Here m1 = 7, m2 = 5, c1 = 7 • (7−1 mod 5) = 7•
3 =, c212 = 5 • (5−1 mod 7) = 5•
3 =, 15so
f −1 ((a1 , a2 )) = 21a1 + 15a2 (mod 35).
(Note: f (21) = , (10), f (15) = , (01).) Thus, f −1 ((1, 5)) = 21 +•
5 15 = 96mod =35). 26 (
Speeding up Modular Exponentation. A significant application is speeding up exponentiation modulo n = pq when p
and q are known. To compute y = xd mod n, where f (x) =x(1 , x2 ):
f (y) = f (xd ) =xd( d
d mod (p−1) mod p, xd mod (q−1) mod q) .
Note xp−1
1 mod p, x2 mod q) =x(1 2
1 = 1 mod p for x1 = 0 by Fermat’s Little Theorem. Then convert back from (y mod p, y mod q) to y mod n. Since exponentiation takes time cubic in the input size, two halfsize exponentiations are about four times faster than one fullsize exponentiation (including conversion).
Exercise 1 Prove that x is a square mod n = pq if and only if it is a square mod p and mod q.
Sunday, May 9, 2010
SYSTEMATIC JEWISH BUSINESS DEVELOPMENT
“ SYSTEMATIC JEWISH BUSINESS DEVELOPMENT ”
8 SALT WISDOM
1 . MENTAL DEVELOPMENT .
Most of the companies or large organizations crash and burn , it happened because of
greed & ego . Other companies did it right. They had good sponsoring systems in place, and
they sold their promotional material at a reasonable price . Network marketing is a teaching &
mentoring business . Your product is people. So study people. Find out how you can help them
reach their dreams . Your #1 goal is to become a lobbyist / mentor with a servant’s heart . The
biggest successes are people-and-relationship-driven companies with good products or services
& good management .
2 . EFFECTIVE MARKETING DEVELOPMENT.
a. marketing plan
b. product knowledge .
c . short and long term evaluation .
3 . FRUGALITY DEVELOPMENT .
4 . SUCCESS SCHEME DEVELOPMENT.
5 . CREATIVITY & FLEXIBILITY DEVELOPMENT.
But between the strategy and the marketing is a gap: the gap between the wisdom of strategy and the effectiveness of marketing. What happens in that gap is magic . Magic is a forbidden topic in business. CEOs routinely consult psychics and astrologers, but they do it covertly, as if public knowl- edge of their interest would tank their stock price and force their resignation. In fact, a passion for alternative knowledge has always gone hand in hand with rigorous rationality. “Even the greatest fig- ures of the scientific revolution dabbled in the mystical arts,” writes Langdon Winner in Autonomous Technology. “Kepler was a confirmed astrologer; Newton tried his hand at alchemy.”
THE FORBIDDEN MAGIC IN OUR AGE ARE CREATIVITY AND FLEXIBILITY, WHICH IS STILL RESERVED FOR ARTISTS AND BOHEMIANS. BUT THIS KIND OF MAGIC IS COMPLETELY CAPABLE OF CARRYING A STRATEGY THAT’S INTERESTING AND RELEVANT AND MEMORABLE TO PEOPLE .
6 . SEARCHING FOR THE FUNDAMENTAL VALUE OVERLOOKED BY THE STOCK MARKET .
7 . NEVER INVESTED UNLESS YOU COULD FIND SOMETHING WORTH BUYING .
8 . NEVER PUT YOUR MONEY INTO ANYTHING YOU DIDN’T UNDERSTAND .
( BE A SUCCESSFUL JEWISH LOBBYIST )
"Where no counsel is, the people fall, but in the multitude of counselors there is safety"
Proverbs XI/14.
8 SALT WISDOM
1 . MENTAL DEVELOPMENT .
Most of the companies or large organizations crash and burn , it happened because of
greed & ego . Other companies did it right. They had good sponsoring systems in place, and
they sold their promotional material at a reasonable price . Network marketing is a teaching &
mentoring business . Your product is people. So study people. Find out how you can help them
reach their dreams . Your #1 goal is to become a lobbyist / mentor with a servant’s heart . The
biggest successes are people-and-relationship-driven companies with good products or services
& good management .
2 . EFFECTIVE MARKETING DEVELOPMENT.
a. marketing plan
b. product knowledge .
c . short and long term evaluation .
3 . FRUGALITY DEVELOPMENT .
4 . SUCCESS SCHEME DEVELOPMENT.
5 . CREATIVITY & FLEXIBILITY DEVELOPMENT.
But between the strategy and the marketing is a gap: the gap between the wisdom of strategy and the effectiveness of marketing. What happens in that gap is magic . Magic is a forbidden topic in business. CEOs routinely consult psychics and astrologers, but they do it covertly, as if public knowl- edge of their interest would tank their stock price and force their resignation. In fact, a passion for alternative knowledge has always gone hand in hand with rigorous rationality. “Even the greatest fig- ures of the scientific revolution dabbled in the mystical arts,” writes Langdon Winner in Autonomous Technology. “Kepler was a confirmed astrologer; Newton tried his hand at alchemy.”
THE FORBIDDEN MAGIC IN OUR AGE ARE CREATIVITY AND FLEXIBILITY, WHICH IS STILL RESERVED FOR ARTISTS AND BOHEMIANS. BUT THIS KIND OF MAGIC IS COMPLETELY CAPABLE OF CARRYING A STRATEGY THAT’S INTERESTING AND RELEVANT AND MEMORABLE TO PEOPLE .
6 . SEARCHING FOR THE FUNDAMENTAL VALUE OVERLOOKED BY THE STOCK MARKET .
7 . NEVER INVESTED UNLESS YOU COULD FIND SOMETHING WORTH BUYING .
8 . NEVER PUT YOUR MONEY INTO ANYTHING YOU DIDN’T UNDERSTAND .
( BE A SUCCESSFUL JEWISH LOBBYIST )
"Where no counsel is, the people fall, but in the multitude of counselors there is safety"
Proverbs XI/14.
Subscribe to:
Posts (Atom)