Volume 12, 2000

University of Bialystok

Copyright (c) 2000 Association of Mizar Users

**Yasushi Fuwa**- Shinshu University, Nagano
**Yoshinori Fujisawa**- Shinshu University, Nagano

- In this article, we propose a new high-speed processing method for encoding and decoding the RSA cryptogram that is a kind of public-key cryptogram. This cryptogram is not only used for encrypting data, but also for such purposes as authentication. However, the encoding and decoding processes take a long time because they require a great deal of calculations. As a result, this cryptogram is not suited for practical use. Until now, we proposed a high-speed algorithm of addition using radix-$2^k$ signed-digit numbers and clarified correctness of it ([6]). In this article, we defined two new operations for a high-speed coding and encoding processes on public-key cryptograms based on radix-$2^k$ signed-digit (SD) numbers. One is calculation of $(a*b)$ mod $c$ ($a,b,c$ are natural numbers). Another one is calculation of $(a^b)$ mod $c$ ($a,b,c$ are natural numbers). Their calculations are realized repetition of addition. We propose a high-speed algorithm of their calculations using proposed addition algorithm and clarify the correctness of them. In the first section, we prepared some useful theorems for natural numbers and integers and so on. In the second section, we proved some properties of addition operation using a radix-$2^k$ SD numbers. In the third section, we defined some functions on the relation between a finite sequence of k-SD and a finite sequence of ${\Bbb N}$ and proved some properties about them. In the fourth section, algorithm of calculation of $(a*b)$ mod $c$ based on radix-$2^k$ SD numbers is proposed and its correctness is clarified. In the last section, algorithm of calculation of $(a^b)$ mod $c$ based on radix-$2^k$ SD numbers is proposed and we clarified its correctness.

- Some Useful Theorems
- Properties of Addition Operation Using Radix-$2^k$ Signed-Digit Numbers
- Definitions on the Relation Between a Finite Sequence of $k$-SD and a Finite Sequence of ${\Bbb N}$ and Some Properties about them
- A High-Speed Algorithm of Calculation of $(a*b)$ mod $b$ Based on Radix-$2^k$ Signed-Digit Numbers and its Correctness
- A High-Speed Algorithm of Calculation of $(a^b)$ mod $b$ Based on a Radix-$2^k$ Signed-Digit Numbers and its Correctness

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