Discrete Logarithm Problem

Discrete logarithms are logarithms defined with regard to multiplicative cyclic groups. If G is a multiplicative cyclic group and g is a generator of G, then from the definition of cyclic groups, we know every element h in G can be written as gx for some x. The discrete logarithm to the base g of h in the group G is defined to be x . For example, if the group is Z5*, and the generator is 2, then the discrete logarithm of 1 is 4 because 24 ≡ 1 mod 5.

The discrete logarithm problem is defined as: given a group G, a generator g of the group and an element h of G, to find the discrete logarithm to the base g of h in the group G. Discrete logarithm problem is not always hard. The hardness of finding discrete logarithms depends on the groups. For example, a popular choice of groups for discrete logarithm based crypto-systems is Zp* where p is a prime number. However, if p−1 is a product of small primes, then the Pohlig–Hellman algorithm can solve the discrete logarithm problem in this group very efficiently. That’s why we always want p to be a safe prime when using Zp* as the basis of discrete logarithm based crypto-systems. A safe prime is a prime number which equals 2q+1 where q is a large prime number. This guarantees that p-1 = 2q has a large prime factor so that the Pohlig–Hellman algorithm cannot solve the discrete logarithm problem easily. Even p is a safe prime, there is a sub-exponential algorithm which is called the index calculus. That means p must be very large (usually at least 1024-bit) to make the crypto-systems safe.

Source: https://www.doc.ic.ac.uk/~mrh/330tutor/ch06s02.html


Cyclic Groups and Generators

Some groups have an interesting property: all the elements in the group can be obtained by repeatedly applying the group operation to a particular group element. If a group has such a property, it is called a cyclic group and the particular group element is called a generator. A trivial example is the group Zn, the additive group of integers modulo n. In Zn, 1 is always a generator:

1 ≡ 1 mod n

1+1 ≡ 2 mod n

1+1+1 ≡ 3 mod n

1+1+1+…+1 ≡ n ≡ 0 mod n

If a group is cyclic, then there may exist multiple generators. For example, we know Z5 is a cyclic group. The element 1 is a generator for sure. And if we take a look at 2, we can find:

2 ≡ 2 mod 5

2+2 ≡ 4 mod 5

2+2+2 ≡ 6 ≡ 1 mod 5

2+2+2+2 ≡ 8 ≡ 3 mod 5

2+2+2+2+2 ≡ 10 ≡ 0 mod 5

So all the group elements {0,1,2,3,4} in Z5 can also be generated by 2. That is to say, 2 is also a generator for the group Z5.

Not every element in a group is a generator. For example, the identity element in a group will never be a generator. No matter how many times you apply the group operator to the identity element, the only element you can yield is the identity element itself. For example, in Zn, 0 is the identity element and 0+0+…+0 ≡ 0 mod n in all cases.

Not every group is cyclic. For example, Zn*, the multiplicative group modulo n, is cyclic if and only if n is 1 or 2 or 4 or pk or 2*pk for an odd prime number p and k ≥ 1. So Z5* must be a cyclic group because 5 is a prime number. Actually all the elements in Z5*, {1,2,3,4} can be generated by 2:

21 ≡ 2 mod 5

22 ≡ 4 mod 5

23 ≡ 8 ≡ 3 mod 5

24 ≡ 16 ≡ 1 mod 5

And Z12* is not a cyclic group. The elements in Z12* are: {1,5,7,11}. Obviously the identity element 1 cannot be a generator. Let’s check the other three elements:

51 ≡ 5 mod 12 71 ≡ 7 mod 12 111 ≡ 11 mod 12
52 ≡25 ≡ 1 mod 12 72 ≡ 49 ≡ 1 mod 12 112 ≡ 121 ≡ 1 mod 12

None of the elements can generate the whole group. Therefore, none of them is a generator. So Z12* is indeed not cyclic.

If Zn* is cyclic and g is a generator of Zn*, then g is also called a primitive root modulo n.

Source: https://www.doc.ic.ac.uk/~mrh/330tutor/ch06.html


[My Escape] – Membuat Resident Visa untuk Studi Lanjut di Taiwan

Mengurus visa merupakan tahapan berikutnya yang harus dilakukan setelah persyaratan sebelumnya, seperti medical check-up, selesai. Gambar di atas merupakan visa yang saya buat untuk keperluan studi lanjut S3 saya di Taiwan. Dapat dilihat di atas, masa berlaku visa tersebut dari waktu dikeluarkannya (Issue Date) hingga masa penggunaannya untuk masuk ke Taiwan (Enter Before) maksimal adalah 3 bulan, dengan kata lain visa tersebut harus digunakan sebelum 3 bulan. Untuk international student kodenya FS (lihat Remarks).

Lama waktu pembuatannya sendiri kurang lebih 2 hari, saya memasukkan berkas pada hari Selasa pagi (kantor Taipei Economic and Trade Office (TETO) buka pukul 08.30 WIB) dan sudah dapat diambil pada hari Kamis (waktu pengambilan visa dilayani mulai pukul 13.30 WIB). Kantor TETO Jakarta (karena ada juga kantor TETO di Surabaya) sendiri beralamat di:
L12 & L17 Gedung Artha Graha, Jalan Jendral Sudirman No. Kav 54-55, Senayan, Kebayoran Baru, RT.5/RW.3, Senayan, Jakarta Selatan, Kota Jakarta Selatan, Daerah Khusus Ibukota Jakarta 12190.
Untuk pengurusan visa sendiri ada di lantai 12, yang nantinya ketika anda masuk gedung tersebut, anda diminta untuk meninggalkan ID card anda dan ditukar dengan kartu akses masuk, serta ditunjukkan oleh petugas security untuk naik menggunakan lift nomor berapa ke lantai 12 tersebut.

Adapun dokumen-dokumen yang perlu anda persiapkan sebelum datang ke kantor TETO di Jakarta untuk mengajukan pembuatan visa antara lain:

1. Mengisi aplikasi permohonan visa secara online melalui website https://visawebapp.boca.gov.tw, kemudian aplikasi permohonan tersebut di-print dan ditandatangani langsung oleh yang bersangkutan.
2. Melampirkan 2 lembar pasfoto 6 bulan terakhir, ukuran 4 cm x 6 cm (berwarna dan latar belakang putih).
3. Paspor berlaku minimal 6 bulan ke atas (Asli + Fotokopi).
4. Surat penerimaan sekolah (LoA) / Admission letter (Asli + Fotokopi). Nb: LoA asli akan ditahan di TETO selama proses pembuatan visa selama +/- 2 hari tersebut.
5. Ijazah + Transkrip nilai terakhir (Bahasa Inggris) (Asli + Fotokopi). Nb: Ijazah + Transkrip nilai asli akan ditahan di TETO selama proses pembuatan visa selama +/- 2 hari tersebut.
6. Jika belum ada Ijazah, lampirkan surat keterangan Lulus (Tidak perlu legalisir, kecuali pihak sekolah yang meminta).
7. Medical Check-up di rumah sakit / klinik yang ditentukan (Asli + Fotokopi)
a. Masa berlaku Medical 3 bulan
b. Jika hasil medical check-up measles dan rubella negatif, maka harus melampirkan bukti vaksinasi.
8. Fotokopi kartu keluarga.
9. Fotokopi rekening tabungan minimal Rp 50.000.000,- dalam 3 bulan terakhir / surat bukti beasiswa. Nb: Dalam case saya, saya hanya menggunakan surat bukti penerimaan beasiswa dari kampus tujuan.
10. Biaya Resident visa Rp. 858.000,- , Biaya Visitor visa Rp 650.000,- . Nb: Pembayaran hanya dapat dilakukan secara tunai/cash.

Nb: Perihal medical check-up

1. Daftar rumah sakit / klinik dapat di download dari website TETO.
2. bagi yang belum menerima Ijazah kelulusan, tidak perlu melampirkan medical dan hanya mendapatkan Visitor visa.
3. TETO berhak untuk meminta penambahan dokumen / panggilan interview.


[My Lecture] – Certificate CCNA 1 S1TT B 2014

Berikut saya lampirkan Certificate CCNA 1 kelas B 2014. Silahkan jika dulu belum diambil / hilang, dsb, dapat dicetak di kertas sertifikat dan dipergunakan sebagaimana mestinya.

Adi_Rahman_Hakim_Suryansyah.2848107.381643
Ardhi_Dwi_Satrio_W.2834047.381643
Bianca_Alfathan_Pratama.2834049.381643
Bimo_Reza_Prayudha.2834050.381643
Bobby_Bayu_Setiawan.2834051.381643
Eka_Larasati_Choerunisa.2834053.381643

Evi_Oktaviasari.2834054.381643
Fildza_Amalia_Zhafira.2834055.381643
Firdha_Amalia_Rakhmat.2834056.381643
Hanang_Adi_Nugroho.2834057.381643
Indra_Nurrahman.2834059.381643
Johannes_Nainggolan.2834061.381643
Juwi_Nanda_Sinulingga.2834062.381643
Mardho_Tillah.2834063.381643
Maria_Chris_Tinna_Tanggahma.2834078.381643
Mauladi_Azhar_M.2834064.381643
Mochamad_Nur_Akbar.2834065.381643
Muhammad_Abojasin_Juliyanto.2834066.381643
Muhammad_Rafi_Raihan.2834068.381643
Nailis_Dyanningrum.2834069.381643
Nizam_Galih_Yudhistira.2834070.381643
Rindu_Wulandari.2834073.381643
Roberto_Pinem.2834074.381643
Syamsul_Huda.2834075.381643
Yogesti_Widyaning_Tyas.2834076.381643
Zuhdi_Alvian.2834077.381643


[My Lecture] – Certificate CCNA 1 S1TT A 2014

Berikut saya lampirkan Certificate CCNA 1 kelas A 2014. Silahkan jika dulu belum diambil / hilang, dsb, dapat dicetak di kertas sertifikat dan dipergunakan sebagaimana mestinya.

Adisti_Nabilah_Naufallia.2833811.381642
Amin_Sudibyo.2837620.381642
Amirul_Hakim_Ardhijanto.2833789.381642
Arfian_Rizki_Wicaksono_.2833787.381642
Eggy_Fauzie_A.2833791.381642
Ervin_Bahar_Panunthun.2833793.381642
Fachry_Rizkyo_Nenfiko.2833817.381642
Irfan_Nur_Aziz.2837621.381642
Ivan_Hertadi_Rahman.2833795.381642
Johanes_Prin_Karo_Sekali.2833797.381642
Jovi_Brema_Barus.2833799.381642
Laela_Amanatul_Fajriyani.2833801.381642
Lina_Azhari.2833803.381642
Mochammad_Muflich_Ashafa.2833815.381642
Muhamad_Syamsul_Fallah.2833813.381642
Muhammad_Al_Balighi_Yuridhol_Furqon.2833809.381642
Muhammad_Iqbal_Al_farisi.2837623.381642
Nova_Ade_Kurniawan.2837622.381642
Nurika_Ayu_Mardaningrum.2833819.381642
Nurul_Haiziah_Nugraha.2833821.381642
Prasojo_Sudiyanto.2833826.381642
Rio_Teguh_Ardiarta.2833824.381642
Rizal_Maulana_Hidayat.2833825.381642
Rut_Elisawanty_Sinaga.2833827.381642
Sri_Utami.2833828.381642
Taufik_Hidayat.2833829.381642
Teguh_Wahyu_Dianto.2833830.381642
Tri_Rahmat_Irianto.2833807.381642
Tri_Retno_Palupi.2833805.381642
Unggul_Riyadi.2833823.381642
Viktor_Purba.2833831.381642
Windyastuti_Herdiningrum.2833832.381642
Yahya_Sosialista_Sembiring_K.2833833.381642
Yanuar_Agung_Firmansyah.2833834.381642


[My Lecture] – Certificate CCNA 2 S1TT A 2014

Berikut saya lampirkan Certificate CCNA 2 kelas A 2014. Silahkan jika dulu belum diambil / hilang, dsb, dapat dicetak di kertas sertifikat dan dipergunakan sebagaimana mestinya.

Amin_Sudibyo.2837620.522283
Amirul_Hakim_Ardhijanto.2833789.522283
Eggy_Fauzie_A.2833791.522283
Ervin_Bahar_Panunthun.2833793.522283

Fachry_Rizkyo_Nenfiko.2833817.522283
Irfan_Nur_Aziz.2837621.522283
Jovi_Brema_Barus.2833799.522283
Lina_Azhari.2833803.522283

Nova_Ade_Kurniawan.2837622.522283
Prasojo_Sudiyanto.2833826.522283
Rizal_Maulana_Hidayat.2833825.522283
Rut_Elisawanty_Sinaga.2833827.522283
Sri_Utami.2833828.522283
Taufik_Hidayat.2833829.522283
Teguh_Wahyu_Dianto.2833830.522283
Tri_Rahmat_Irianto.2833807.522283
Unggul_Riyadi.2833823.522283

Viktor_Purba.2833831.522283
Windyastuti_Herdiningrum.2833832.522283
Yanuar_Agung_Firmansyah.2833834.522283


[My Lecture] – Komunikasi Serat Optik – Tugas 2

Tugas Komunikasi Serat Optik ini dikumpulkan saat UAS Komunikasi Serat Optik, 22 Januari 2018, pukul 18.30 WIB.
*Kerjakan di kertas folio bergaris.

Tugas 2


[Scholarship] – Taiwan Higher Education Fair 2018


[Article] – OSI: The Internet That Wasn’t

“How TCP/IP Eclipsed the Open Systems Interconnection Standards to Become the Global Protocol for Computer Networking”

Sumber: IEEE Spectrum, The Internet That Wasn’t


[My Lecture] – Referensi Tambahan Rekayasa Trafik (MKB2333)

Berikut adalah referensi untuk MK Rekayasa Trafik (MKB2333) TA Ganjil 2017/2018:

Referensi Utama:

Rekayasa Trafik Telekomunikasi, Sofia Naning H.
Teletraffic Enginering and Network Planning, Villy B. Iversen
Tabel Erlang

Referensi Tambahan:
Traffic Theory and the Internet

Link:
https://goo.gl/LRHjoR