RESIDUE NUMBER SYSTEM. (introduction to hardware aspects) Dr. Danila Gorodecky

Transcription

1 RESIDUE NUMBER SYSTEM (introduction to hardware asects) Dr. Danila Gorodecky

2 Terminology Residue number system (RNS) (refers to Chinese remainder theorem) Residue numeral system (RNS) Modular arithmetic (MA) (refers to moduli X (mod P) ) Comlete residue system Clock arithmetic (refers to -hour arrow clock in which numbers "wra around uon reaching the modulo)

3 Milestones Chinese mathematician Sunzi Suanjing roosed a theorem (Chinese remainder theorem) in the 3rd century AD; the theorem was generalized by Chinese mathematician Qin Jiushao in 47; first real imlementation of the theorem by German mathematician Carl Gauss in 80 "to find the years that have a certain eriod number with resect to the solar and lunar cycle and the Roman indiction ; first imlementation in comuter science by Czechoslovakian engineer Miro Valach in 955 Origin of the code and number system of remainder classes, Stroje Na ZracovaniInformaci, vol. 3, Nakl. CSAV, Prague.

4 Imlementation of RNS Processing of results of the Unified State Exam (utilized to entrance to University in Russia; Digital filternig with finite imulse resonse (FIR-filtering); Cryto system of Federal Reserve System of USA; Air Defense System (USA, Russia); crytograhy in Sace (Russia); Sace flight control (Russia) 4

5 Chinese remainder theorem Let s,,..., n are ositive integers (are often called as moduli) such, that greatest common divisor for a coule i j equals. Then the system, yx yx... mod mod yx n mod n has a simultaneous solution which is unique modulo,..., n, 5

6 Examle for the Theorem P We can exress an arbitrary number definitely in the scoe from 0 to 34 Let s A00, hence A000 A00 A00 mod 5 mod 7 mod 9 and A 0,, in the RNS reresentation 6

7 NOT ositional numeral system RNS is not ositional numeral system 00(mod 5), 00(mod 7), 00(mod 9) (0,, ) 00 00(mod 7), 00(mod 9), 00(mod 5) (,, 0) 00 Binary system (0000) (0000)

8 Examle of the comutation in RNS P AB003S 35 ) A0,, B3,6,4 ) A B 0 3 mod 5, 6 mod 7, 4 mod 9 3 mod 5, mod 7, 5 mod 9 3,,5 8

9 Y i 3) Examle of the comutation in RNS S SY S Y S3 Y3 r P P ki ; Y i (mod i ); i i r P S Y S Y S Y r P 3 3 a) 35 Y k 63 k 5 35 b) Y k 45 k 7 35 с) Y3 k3 35 k k 5 (mod 5) and, then k and Y 6 k and, then and Y 5 45 (mod 7 35 k 3 (mod 9 7) 9) k 5 and, then k 3 8 andy 3 80 d) 35 r r r, then 6 S

10 Exercises ) What is maximum bit range of A and B should be chosen for unambiguous reresentation A + B = S in RNS with moduli,3, and 5? P = * 3 * 5 = 45 0 S < 45 and S is -bit number. Hence, in order to reresent A + B = S, A and B should be limited -bit tules, when A and B both equal 077. ) What is maximum bit range of A, B, and C can be used for unambiguous reresentation A * B * C = R in RNS with moduli,3, and 5? P = * 3 * 5 = 45 0 S < 45 and S is -bit number. Hence, in order to reresent A * B * C = R, A, B, and C should be limited 4-bit tules, when A, B, and C equal. 0

11 Main advantage of RNS Significantly smaller ranges of numbers in arithmetic calculations than initial numbers

12 Examle of the comutation in RNS A B i i A { A, A, A3} (mod i ) A 0A34 A i B { B, B, B3} B (mod i ) B i 0B34 i i SSY SY S3Y 3rP Yi (mod i ) i Yi P i k i i,,3 k i, i r 0,,,...

13 X (mod P) hardware calculations in RNS X mod P A A Converter of A A g summator/multilier (mod ) S Converter of A ositional numbers to A A A g summator/multilier (mod ) S modular reresentation to S A g modular reresentation () A A summator/multilier (mod m ) S m ositional number () A g 3

14 4 Pielining aroach of X (mod P) calculations ) Pielining (iterative) aroach. It is based on the formula: A q P q P q P A Q P X ,,, x x x X...,,, P P P

15 50 Pielining aroach of X (mod P) calculations mod7 7Q Amod q6 7 q5 7 q4 7 q3 7 q 7 q 7 q0 Amod 7 Amod mod 7 7Q A 7 7 Amod 76 mod 7 5

16 Recursive aroaches of calculation X (mod P) in hardware ) Recursive aroach. It is suitable for secial moduli, e.c. and X as x x x3... x x x, x,..., x X x, x,..., x X X... X X X3... n x, x,..., n 3 x,..., so X X X X... mod mod mod 3 {0, } Examle. X (mod P), where and X P 3 7 a) mod7 000 mod 7 b) mod 7 0 mod 7 3 mod 7 6

17 Recursive aroaches of calculation X (mod P) in hardware 3) It is suitable for an arbitrary modulo and is based on the next formula: X x X x x x3 x4... mod P 3 x mod P x mod P x mod P x mod P... Examle. X (mod P), where 0 and 9x If If 3 X x, x,..., x P3 4 x, x, x3, x4,... {0,} x x x3 x4 x5 x6 x7 x8 x9 x0 mod3 mod3 xmod34 x3mod38 x4mod36x5 mod3 mod38x mod33x mod33 x mod36 x mod3 6 0S3 3 Let s 7 X mod X mod X P 3S46 X 3 8 If If 9 0 mod 3 X P mod 3 X P 46S69 X 69S9 X 3 mod 3 X 030 (mod 3) 7

18 ^n (mod P) calculation in hardware 4) X (mod P), where n P n n x, x,..., x, x,..., x mod X mod n n n x x,..., x mod n, n x, x, x3, x4,... {0,} X P6 Examle. X (mod P), where and X mod

19 Exercises Using one of the considered techniques, calculate: ) (mod 3 ) 0? ) (mod 3 )? 3) ( ) (mod 3 )? ( ) (mod )? 4) with technique 3) 3

20 Arithmetic calculations in RNS Arithmetic calculations on moduli A A Converter of A A g summator/multilier (mod ) S Converter of A ositional numbers to A A A g summator/multilier (mod ) S modular reresentation to S A g modular reresentation () A A summator/multilier (mod m ) S m ositional number () A g 0

21 Arithmetic calculations in RNS Standard aroach of arithmetic calculations in RNS includes ) arithmetic calculations (A B=R, A+B=S, and etc., where A and B vary from 0 to P-); ) modulus function calculation (R (mod P), S (mod P), and etc.) Examle. A B=R (mod 7), hence A and B vary from 0 to 6. Lets A=5 and B=6. ) ) (mod (00) 7) (0) ((0) (0)) (mod ((00) (00)) (mod 7) (mod 0 (mod 7) 7) 7) (mod (00)(mod 7) 7)

22 Arithmetic calculations in RNS 300 What is about P? ) A B R 600 ) R (mod 300 ) or Montgomery and a-la Montgomery multilication: Examle. (56) (mod 7) ((0) 4 (0) (mod 7) )(mod 7) (mod 7) (( 9(mod 7) (00)(mod 7) 3 (mod 7) 3 0 ) ( (mod 7) ))(mod 7) (mod 7) ( 4 )(mod 7) 0 (mod 7) ( )(mod 7)

23 Exercises ) How many rows and columns in the truth table of system of Boolean functions, which reresents A + B = R (mod 5)? ) How many rows and columns in the truth table of system of Boolean functions, which reresents A * B = R (mod 7)?

24 Backward conversion S S Y S Y... S m Y m r P A A Converter of A A g summator/multilier (mod ) S Converter of A ositional numbers to A A A g summator/multilier (mod ) S modular reresentation to S A g modular reresentation () A A summator/multilier (mod m ) S m ositional number () A g 4

25 S S Y S Y... Yi (mod i ) i S Y i m Y P Backward conversion i m k i r P ) multilication by a big number; ) big numbers summation; 3) comarison Examle. A B in RNS with moduli set P,, } {3,3,33} { 3 R s 6864 s 373 s r

26 Exercises Multily 9 * 03 in RNS with moduli set {, 3, 5, 6}?

27 Electronic Design Automation (EDA) tools Synosys executes X mod P; Xilinx (ISE, Vivado) imlementation IPblocks; LeonardoSectrum (Mentor Grahics) allows to use custom libraries; and etc. 7

28 Secial sets of moduli alied in RNS 8

29 Five moduli set A B A, B In order to calculate, where, 739 the average bit-range of 5 moduli sets is 300 bits, i.e

30 Moduli set for A B, where A and B are 739 bits P = { } 7 P 478 P 30

31 Features and roblems It is assumed, that: the main feature is the high seed rocessing (it is achieved with hundreds bits numbers); indeendence of calculation under each modulo; flexibility of layout; small ower consumtion; reliability Problems: unknown an efficient aroach of hardware realization for an arbitrary modulo P no IP-blocks and no hardware libraries for RNS system realization; slow seed converters to/from RNS for non secial sets of moduli 3

32 Residue Number Systems: Algorithms and Architectures Kluwer Academic Publishers, 00 RNS literature (in English) Digital arithmetic Morgan Kaufmann Publishers, 004 Residue Number Systems: Theory and Imlementation Imerial College Press, 007 Residue Number System Bookvika Publishing, 0 Finite recision number systems and arithmetic Cambridge University Press, 00 3