## Drawing Elliptic Curve with Geometer Sketchpad

Point Doubling:

Associative:

[*Note: Big thanks for Fajar Yuliawan for his brilliant tutorial 😉 ]

• #### Budi Rahardjo 2:39 pm on November 4, 2011 Permalink | Reply

yay! good stuff …

• #### CG 7:41 am on August 14, 2010 Permalink | Reply Tags: digital signature, elliptic curve, hardware design ( 3 ), vhdl ( 45 )

A Low-Power VHDL Design for an Elliptic Curve Digital Signature Chip – Richard Schroeppel, Cheryl Beaver, and Timothy Draelos – Cryptography and Information Systems Surety Department, September 2002

## Doubling & Addition Table (4 bits)

Notes:

1. The calculation starts from LSB
2. Every bit shift means doubling, bit 1 means added by P and bit 0 means not added by P
3. Adding costs more than doubling

• #### Budi Rahardjo 5:19 pm on February 13, 2010 Permalink | Reply

to be frankly, i cannot see the pattern from the table.
i have different notation in my head 🙂 which is basically similar to what you wrote in your notes.

• #### CG 5:42 pm on February 13, 2010 Permalink | Reply

@BR: should there be any pattern anyway? 😛 i’m so lost in multiplication of different bases 😀 show me your notation 🙂

## Newcomers to the bookshelf :)

Forgot to post an update about me shopping some books, really cool ones 🙂

2. A very detail and theoritical book about elliptic curves: “Elliptic Curves – Number Theory and Cryptography” – Lawrence C. Washington

3. The most related book to be the reference of my phd thesis: “Elliptic Curve Cryptography for Constrained Devices – Algorithms, Architectures and Practical Implementations” – Sandeep S. Kumar

Happy holiday, everybody.

I’m going to spend the holiday with my new buddies 😉

• #### Budi Rahardjo 7:13 pm on December 26, 2008 Permalink | Reply

Waaahhh … iri …
Buku baru selalu membuat iri 😀

• #### CG 7:15 pm on December 26, 2008 Permalink | Reply

@BR: hey, you’ve got to read them too and help me to understand and finish my phd 😀

• #### mehobbes 5:39 am on December 29, 2008 Permalink | Reply

the first one is a “bible” ?
hmmmm, oke, a must have ECC book isn’t, it.

## Changing generators

I was going to observe the “behaviour” of an elliptic curve by changing its generator, and looking for an answer what does happen if I change it.

From the discussion yesterday, I understand that each generator will generate different cyclic subgroups. And does it have something to do with security level? Let’s find out.

Still thinking about changing other parameters of elliptic curve, and observe the result.

## Why finite fields?

Eh, there are still some questions left, and I’m posting it here to remind me that I have to move forward from this point 😉

Question:

Why finite fields? Does it has something to do with “reversible”? Is that a requirement for only elliptic curve? Is it possible for elliptic curve without finite fields?

## Order of the point

Multiplication over elliptic curve is like this:

$Q = kP$

where $Q$ and $P$ are points on an elliptic curve and $k$ is an integer. The equation above means that $P$ is added to itself $k$ times.

What I don’t understand (yet) is, why the integer $k$ need not be larger than the “order” of the point $P$ ? Understand that reducing $k$ will save a great deal of processing. But what’s the relation between the “order” of the point with this statement:

The points in elliptic curve, forms a cyclic group (a field)

Update:

I guess now I start to grab the ideas about those previous paragraphs 🙂

I’ll check it to mathematician buddy as soon as possible 😉

• #### tetangga sebelah 7:03 pm on October 6, 2008 Permalink | Reply

walah baca postingan ini jadi merasa orang garut……dodol
cing atuh……pake bahasa manusia kalau posting tehhhh

@tetangga sebelah: cing atuh karunya ka sayah. lebaran mudik teh bawain oleh2x atuh, opor kek, dodol kek…

emang bukan bahasa manusia makanya saya juga ga ngarti…

• #### gre 5:51 pm on October 10, 2008 Permalink | Reply

speaking of field, you must be a knowledgeable person in abstract algebra and it is rarely found in LABTEK VIII 🙂
I am just wondering if the k integer might be a Lipshitz constant.

@gre: not yet a knowledgeable person in abstract algebra 😉
what is a Lipshitz constant anyway? [pls don’t give more headaches to me now 😀 ]

## Applying plaintext embedding

Continuing this, from “Problems of Plaintext Embedding on Elliptic Curve” by Fucai Zhou and Jun Zhang, I’m now focusing on the algorithms of plaintext embedding on Elliptic Curve.

Will explore this more, while also learn about affine, standard and jacobi projective.

## Papers on ECC Implementation

1. D. Hankerson, J.L. Hernandez, A. Menezes, “Software Implementation of Elliptic Curve Cryptography over Binary Fields”
2. M. Brown, D. Hankerson, J. Lopez, A. Menezes, “Software Implementation of the NIST Elliptic Curves over Prime Fields”

• #### Budi Rahardjo 5:20 am on August 29, 2008 Permalink | Reply

Also, don’t forget to enter this into you bibliography database.

• #### intan 1:53 pm on September 3, 2008 Permalink | Reply

@intan: those are papers, not books. would be glad to give presentation, but need some time to prepare it. is 2 weeks from now ok? 😉

## Plaintext embedding on Elliptic Curve: Conclusions

From this paper:

• Problems of plaintext embedding are the fundamental problems in elliptic curve public key cryptosystems.
• A good embedding algorithm will enhance the encryption speed.
• Plaintext embedding algorithm in binary field presented in the paper is easier to be implemented than that in prime field, and faster.
• Applying the plaintext embedding algorithm to the storage and trasmission of points in elliptic curve can save half of the storage space and bandwidth.

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