Numeration and Computer Arithmetic Some Examples

Transcription

1 Numeration and Computer Arithmetic 1/31 Numeration and Computer Arithmetic Some Examples JC Bajard LIRMM, CNRS UM2 161 rue Ada, Montpellier cedex 5, France April 27

2 Numeration and Computer Arithmetic 2/31 Computer Arithmetic Compromise: Speed Accuracy Cost Heart: Number representations Associated algorithms Approaches: Theory Software Hardware

3 Numeration and Computer Arithmetic 3/31 Contents Function Evaluation Redundant Number Systems Number Systems for Modular Arithmetic Conclusion Annexes

4 Numeration and Computer Arithmetic 4/31 Function Evaluation Function Evaluation an example of numeration

5 Numeration and Computer Arithmetic 5/31 Function Evaluation Briggs Algorithm ( ) Evaluation of the logarithm, constructions of the first tables (15 decimal digits, 1624). In radix 2: digits d k = 1,, 1, such that for a given x we have n x (1 + d k 2 k ) 1 The logarithm of x is k=1 ln(x) n ln(1 + d k 2 k ) k=1

6 Numeration and Computer Arithmetic 6/31 Function Evaluation CORDIC Algorithm (COrdinate Rotation DIgital Computer, VOLDER 1959) Basic step d n { 1, 1} (sign of z). x n+1 = x n d n y n 2 n y n+1 = y n + d n x n 2 n z n+1 = z n d n arctan(2 n ) K sinθ θ K cos θ For cosine and sine: x = 1, y =, z = θ(= n d n arctan(2 n )) Constant factor K = n= n =

7 Numeration and Computer Arithmetic 7/31 Function Evaluation Complex algorithm (BKM 1993) Basic step of the complex algorithm: { Ek+1 = E k (1 + d k 2 k ) L k+1 = L k ln(1 + d k 2 k ) with d k = dk r + id k i, and d k r, d k i = 1,, 1. Two evaluation modes L-mode : E n 1 L n L 1 + ln(e 1 ) E-mode : L n E n E 1 e L 1

8 Numeration and Computer Arithmetic 8/31 Function Evaluation 1 E 1 = n (1 + d i 2 i ) L n = i=1 n ln(1 + d i 2 i ) = ln(e 1 ) i=1

9 Numeration and Computer Arithmetic 9/31 Redundant Number Systems Redundant Number Systems

10 Numeration and Computer Arithmetic 1/31 Redundant Number Systems Avizienis (1961) Redundant Number Systems Signed digits: x i { a,..., 1,, 1,..., a} Radix β with a β 1. Properties If 2a + 1 β, then each integer has at least one representation. An integer X, with a βn 1 β 1 X < a βn 1 β 1, admits a unique representation n 1 X = x i β i with x i { a, 1,, 1,..., a} i= If 2a β + 1, then we have a carry free algorithm. 25 Borrow-save (Duprat, Muller 1989): extension to radix 2.

11 Numeration and Computer Arithmetic 11/31 Redundant Number Systems Example: radix 1, a = (= ) (= 4647) 1111 (= t) 2711 (= w) (= s = 11891)

12 Numeration and Computer Arithmetic 12/31 Redundant Number Systems Properties of the signed digits redundant systems Advantages: Constant time carry-free addition Large radix: parallelisation Small radix: fast circuits Increasing of the performances of the algorithms based on the addition 7 Drawbacks: comparisons, sign...

13 Numeration and Computer Arithmetic 13/31 Redundant Number Systems Non-Adjacent Form This representation is inspired from Booth recoding (1951) used in multipliers. Definition of NAF w recoding: (Reitwiesner 196) Let k be an integer and w 2. The non-adjacent form of weight w of l 1 k is given by k = k i 2 i where k i < 2 w 1, k l 1 and i= each w-bit word contains at most one non-zero digit. 1. For a given k, NAF w (k) is unique. 2. For a given w 2, the length of NAF w (k) is at most equal to the length of k plus one. 3. The average density of non-zero digits is 1/(w + 1)

14 Numeration and Computer Arithmetic 14/31 Redundant Number Systems NAF w Examples We consider k = k 2 = NAF 2 (k) = NAF 3 (k) = NAF 4 (k) = NAF 5 (k) = NAF 6 (k) =

15 Numeration and Computer Arithmetic 15/31 Number Systems for Modular Arithmetic Number Systems for Modular Arithmetic

16 Numeration and Computer Arithmetic 16/31 Number Systems for Modular Arithmetic Lattices and Modular Systems Number system: radix β and a set of digits {,..., β 1}. n 1 A < β n is expanded as: A = a i β i. i= We denote by P the modulo, with P < β n, n 1 β n (mod P) = ɛ i β i with ɛ i {,..., β 1} i= A modular operation (for example: a modular multiplication): 1. Polynomial operation: W (X ) = A(X ) B(X ) n 1 2. Polynomial reduction : V (X ) = W (X ) mod (X n ɛ i X i ) 3. Coefficient reduction : M(X ) = Reductcoeff(V (X )) i=

17 Numeration and Computer Arithmetic 17/31 Number Systems for Modular Arithmetic Lattices and Modular Systems Lattice approach In a classical system Reductcoeff is equivalent to a combination of the carry propagation and the modular reduction: β 1... β β 1 P... lattice sublattice β 1... β β 1 ɛ ɛ 1... ɛ n 2 (ɛ n 1 β)

18 Numeration and Computer Arithmetic 18/31 Number Systems for Modular Arithmetic Lattices and Modular Systems Example For P = 97 and β = 1, we have (mod P). We consider the lattice: ( ) ( ) B 1 1 = 3 1 B 1 Let V (25, 12) = β. For reducing V, we determine G(17, 8) = 2B B 1 a vector of the lattice close to V. Thus, V (25, 12) M(8, 4) = V (25, 12) G(17, 8). We verify that = (mod 97)

19 Numeration and Computer Arithmetic 19/31 Number Systems for Modular Arithmetic Lattices and Modular Systems Example The reduction is equivalent with finding a close vector. Let G(X ) be this vector, then M(X ) = V (x) G(X ) V(25,12) G M P = 97 β = 1

20 Numeration and Computer Arithmetic 2/31 Number Systems for Modular Arithmetic Lattices and Modular Systems A new system Polynomial reduction depends of the representation of β n (mod P) In Thomas Plantard s PhD (25), β can be as large as P, but with a set of digits {,..., ρ 1} where ρ is small. Example: Let us consider a MNS defined with P = 17, n = 3, β = 7, ρ = 2. Over this system, we represent the elements of Z 17 as polynomials in β, of degree at most 2, with coefficients in { 1,, 1}

21 Numeration and Computer Arithmetic 21/31 Number Systems for Modular Arithmetic Lattices and Modular Systems A new system β 2 1 β β + β 2 β + β β β 1 + β 1 β β 1 β β β 2 1 β β β 2 β β 2 The system is clearly redundant. For example: 6 = 1 + β + β 2 = 1 + β, or 9 = 1 β + β 2 = 1 β.

22 Numeration and Computer Arithmetic 22/31 Number Systems for Modular Arithmetic Lattices and Modular Systems Construction of Plantard Systems In a first approach, n and ρ = 2 k are fixed. The lattice is constructed from the representation of ρ in the number system. P and β are deduced. Efficient algorithm for finding a close vector. 31 In a general approach, where P, β and n are given, the determination of ρ is obtained by reducing with LLL (Lenstra Lenstra Lovasz, 1982). No efficient algorithm for finding a close vector. 29

23 Numeration and Computer Arithmetic 23/31 Conclusion Conclusion Thank you!

24 Numeration and Computer Arithmetic 24/31 Annexes Annexes

25 Numeration and Computer Arithmetic 25/31 Annexes Annexe: Avizienis Algorithm 1 We note S = X + Y with X = x n 1...x Y = y n 1...y S = s n...s Step 1: For i = 1 to n in parallel, t i+1 = 1 if, x i + y i < a if, x i + y i > a 1 if, a + 1 x i + y i a 1 and w i = x i + y i β t i+1 with w n = t = Step 2: for i = to n in parallel, s i = w i + t i

26 Numeration and Computer Arithmetic 26/31 Annexes Annexe: Functions computable using one mode of BKM 7 b E x 1 E x n be a 1 cos θ a E x 1 E x n E x 1 E x n c a E y 1 L x 1 E-mode En y L x n E y 1 L x 1 E-mode En y L x n sin θ b E y 1 L x 1 E-mode En y L x n d L y 1 L y n θ L y 1 L y n θ L y 1 L y n a E x 1 E x n 1 a E x 1 E x n 1 E y 1 L x 1 L-mode En y L x n ln a b E y 1 L x 1 L-mode En y L x n ln a 2 + b 2 L y 1 L y n L y 1 L y n arctan b a ( d c ) = ( ) cos θ sin θ sin θ cos θ ( a b )

27 Numeration and Computer Arithmetic 27/31 Annexes Annexe: NAF w Computing 13 Data: Two integers k and w 2. Result: NAF w (k) = (k l 1 k l 2... k 1 k ). l ; while k 1 do if k is odd then k l k mod 2 w ; if k l > 2 w 1 then k l k l 2 w ; end k k k l ; else k l ; end k k/2, l l + 1; end

28 Numeration and Computer Arithmetic 28/31 Annexes Annexe: Double and Add with NAF w 13 Data: P E, k = N et w 2, NAF w (k) = (k l 1 k l 2... k 1 k ) P i = [i]p pour i {1, 3, 5,..., 2 w 1 1} Result: Q = [k]p E. begin Q P kl 1 ; pour i = l 2... faire Q [2]Q; si k i alors si k i > alors Q Q + P ki ; sinon Q Q P ki fin fin fin end

29 Numeration and Computer Arithmetic 29/31 Annexes Lattices and Modular Systems Annexe: Examples of Plantard System 22 Example1: P = 53, n = 7, β = 14, ρ = 2. We have β 7 2 (mod P). In this number system, integers have at least two representations, the total number of representations is 128. The lattice could be defined by (vectors in row): V 1 V 2 V 3 V 4 V 5 V 6 V 7 =

30 Numeration and Computer Arithmetic 3/31 Annexes Lattices and Modular Systems Annexe: Examples of Plantard System 22 We can remark that there is a short vector : (1, 1,,,,, 1) = V V V V V 2 + ( ) V 1 + V 7. From this vector we can construct a reduced basis of a sublattice, using that: β 7 2 (mod P)

31 Numeration and Computer Arithmetic 31/31 Annexes Lattices and Modular Systems Annexe: Examples of Plantard System 22 Example #2: This example is proposed in PhD of Thomas Plantard. He gives some conditions that number system must verify: β 8 2 (mod P) and ρ = P is the determined: P = Then β is deduced β =