Representing integer multiplication using BDDs Hvard Raddum and - PowerPoint PPT Presentation

Representing integer multiplication using BDDs Håvard Raddum and Srimathi Varadharajan Simula@UiB

What is a BDD? Can be defined and understood in different ways Only one description given here

BDD explained A BDD is a directed acyclic graph, drawn from top to bottom 2 1 3 4 6 5

BDD explained Every node can have two outgoing edges, the 0-edge and the 1-edge 1 e 1 g - d e e d - g 0 e 3 2

BDD explained Nodes are arranged in horizontal levels Level i 1 2 3 Level i+1 4 5 Level i+2 6 7 8 9 No edges between nodes on same level

BDD explained Only one node on highest and lowest levels Top node, source node Bottom node, sink node T

BDD explained Linear combination of variables associated with each level, except bottom level x 1 +x 5 1 2 3 x 2 4 5 x 0 +x 2 +x 3 +x 6 6 7 8 9 Other descriptions: only single variables associated with levels

BDD Examples

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Paths 0 1 0 1 0 0 0 0 1 0 1 0 0 1 1 0

Assigning values Values in path assigned to linear combinations associated with levels ��� x 0 = 0 � x 1 + x 2 = 1 ������� � � x 0 + x 2 + x 3 = 0 ����������� � � x 2 + x 3 = 0 ������� � � �

Integer multiplication as BDD How to construct multiplication BDD?

N, p and q in binary • RSA modulus N=pq • p=p n-1 p n-2 … p 2 p 1 p 0 • q=q n-1 q n-2 … q 2 q 1 q 0 • N=N 2n-1 N 2n-2 … N 2 N 1 N 0 • N i ,p i ,q i ∈ {0,1}

Multiplication n n-1 n+1 2 1 0 2n-2 columns Column k consists of all terms p i q j where i+j=k Column k contributes 2 k ∑ p i q j to the value of N

Building multiplication BDD • BDD built column by column, starting with column 0 • One level of nodes constructed from each variable p i , q j appearing in column k p i q j Basic building block: p i q j =1 p i q j =0

Column 0 N=…011 p 0 p 0 q 0 q 0 1 0 1 p 0 q 0 =0 p 0 q 0 =1 When construction of column k keep track of is complete, delete nodes with value of N values inconsistent with N k computed so far

Column 1 N=…011 p 0 p 0 col 0 col 0 q 0 q 0 p 0 p 0 1 1 q 1 q 1 col 1 col 1 p 1 p 1 1 3 1 3 q 0 q 0 3 1 3 5 col 2 col 2

Column 2 N=…011 p 0 p 0 3 3 q 2 q 2 p 1 p 1 3 7 3 7 col 2 col 2 q 1 q 1 p 2 p 2 3 7 11 3 7 11 q 0 q 0 3 7 3 11 15 11 col 3 col 3

Column k col k-1 v v+2 k v+t*2 k p i q j v v+2*2 k v+2 k v+(t+1)*2 k v+t*2 k col k v v+2*2 k v+2 k v+s*2 k v+(s-1)*2 k col k+1

Column 2n-2 col 2n-3 v v+2 2n-2 v+t*2 2n-2 p n-1 col 2n-2 q n-1 v v+2*2 2n-2 v+2 2n-2 v+(t+1)*2 2n-2 v+t*2 2n-2 Only keep node with value N. This becomes the bottom node of the BDD

BDD for multiplication of two 8-bit numbers N=31439 891 nodes > 2 110 paths

General properties of multiplication-BDD • BDD for n-bit numbers p and q has: • 2n different variables • 2n 2 levels (each variable occurring n times) • ≤ 2n 3 - 9/2n + 5 nodes

Using BDD for factoring How to find values of p and q?

Finding lsb’s of p and q p 0 • When N is odd, top of BDD will always be q 0 1 The two top levels give p 0 =q 0 =1 Fix p 0 =q 0 =1 in whole BDD by deleting 0-edges going out from levels with p 0 or q 0

Finding lsb’s of p and q Gives p 1 =q 1 p 1 q 1 Replace q 1 with p 1 on all levels p 2 p 1 q 1 q 2 Depending on value of N, some more linear equations can be extracted from top of BDD

Swapping levels • There exists an algorithm for swapping adjacent levels x y x y • Number of nodes may increase on affected level

Absorbing dependency • Two adjacent levels with equal variables can be merged x x x

Factoring algorithm • Repeat until each p i , q j only occurs once in whole BDD: • Swap levels repeatedly such that two adjacent levels get equal variables • Merge levels • Any path in resulting BDD gives values of p i and q j such that pq=N

p 0 q 0 q 1 Only two paths remain p 1 q 2 p=211 and q=149 p 2 q 3 or p 3 q 4 p 4 p=149 and q=211 q 5 p 5 q 6 Number of nodes in p 6 BDD was never above q 7 891 during solving p 7

Factoring larger numbers? • Experiments show that simple swap-and-merge algorithm has complexity ≈ √ N ≈ 2 n , measured as maximum number of nodes in BDD during solving • However, BDD representation allows to try other tricks

All levels with p 4 merged and moved to the top p 4 p 4

q 7 q 6 p 5 p 0 q 5 p 6 q 0 q 4 p 7 q 3 p 5 q 6 All levels with q 7 p 6 merged and q 5 p 7 moved to the top q 4 p 6 q 6 p 7 q 5 q 7 =0 ⇓ p 7 q 6 q 5 =1 q 4 =1 q 3 =1 q 6 =1 p 6 =1 p 7 =1 p 5 =1