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1 //6 JANT

2

3 98 A.K.estra H.W.estraJr..ovasz lattice asis reduce NT ( Factorig Polyomials with ratioal coefficiets Math.A

4 (lattice } { ( i B z z z z K K Z R det( ( B d G U UB B ( ( ' ( ( ' ( ( B d B d B B

5 B B ( ( ( ( B d B d B B ( B d

6 (shortest vector prolem / λ ( d( (closest vector prolem (smallest asis prolem

7 Gram-Schmidt ( ( * * * * * j j j i ij i j j ij i i µ µ ( (-reduced asis i i j i ii i ij < < 4 3 * * µ µ ( (

8 ( ( ( (4 ( (3 for ma ( ( 4 / ( / 4 / ( / ( / ( d d d t j i i t t j t K K K (estra estra ovasz 98

9 : K ( ( k K (4 k k ( ( (3 K k k ( k k ( k k k ( 98 ( 4 O log B B ma i

10 B ( 5.87 ( ( 4 / 4 / (4 4 / B d B d B d ' 5.9 ' ' ' ' 4 3 B

11 #iclude <NT/.h> mai( { } NT mat_zz ; ci >> ; _FP(; cout << << " "; widows/ui (GP C GMP [ [ ] [ ] [ ] [ ] ] [[ ] [ ] [ ] [ ] ]

12 Geometries of umers SVP CVP SBP lattice Ajtai-Dwork C.P.Schorr Blockwise Korkie-Zolotarev segmet Kaa

13 SVP NP Ajtai STOC 998 SVP f( < / NP Micciacio FOCS 998 (/O(log / NP Goldreich Goldwasser STOC 998 SVP CVP SVP NP P. va Emde Boas 98 Ajtai Average case worst case c -SVP 996 (-/

14 c -SVP worst case average case reductio P NP worst case average case Ajtai-Dwork STOC 997 Nguye Ster C98 3 M impractical

15 ^O((loglog /log Schorr 987 O(^ Kaa STOC 983 serial BKZ Schorr(994 C95 segmet Schorr( Calc parallel Wetzel (ANTS III 998 ad hoc deep isertio Cohe

16 segmet C.P.Schorr (Calc O( 3 log (? (8MHz PC

17 lattice

18 8 Brickell C83 Metal Poker Shamir 79 Coppersmith C84 ow epoet RSA Hastad C85 SIAM J.Comp 88 agarias JACM85 ESIGN Vallee E88 e Vallee AAEEC-6 88 trucated cogruece geerator Frieze SIAM J.Comp88 Isselhorst E89 Ster E9 modular kapsack Niemi E9 Chee C9Jou A9 Coster E9 Diophatie Schorr E kapsack hash Damgard C89 Jou E94 Numer Field Sieve Motgomery Math.Comp.94

19 Chor-Rivest Chor IT88 94 Chor-Rivest Chor IT88 Schorr E95 Coppersmith E96 JC97 Coppersmith E96 JC97 Diffie-Hellma Boeh C96 Hastad oud 96 Buchma C97 ISO/IEC Misarsky C97 Qu-Vastoe Nguye C97 NTRU (Hoffstei C96 Coppersmith E97 Coppersmith Howgrave-Graham CCS97 Jaco Buhler 97 Numer Field Sieve Nguye ANTS III 98 Ajtai-Dwork Nguye C98

20 Julta E98 SAC97 Nguye SAC98 Np r q Boeh C99 GGH Goldreich C97 Nguye C99 Hidde suset sum Nguye C99 d<n.9 RSA Boeh E99 - ISEC99 Noisy Polyomial Bleicheacher E Boeh STOC two-third method Vallee Math.Comp.??

21 agarias Brickell Coppersmith Hastad Hastad Vallee d<n.9 RSA Np r q Howgrave-Graham attice Hidde suset sum DH

22 aa a m m {} am am C aa a C a a C {}

23 a a C ( a ( a ( a ( C ( a ac - a a C ( { }

24 / (ma ai < > {} {} SVP <.6463 agarias CVP <.948 Coster

25 NTRU ( NTRU J.Hoffstei J.Pipher J.H.Silverma 996 lattice reductio(coppersmithshamir (

26 NTRU ( (mod (3 mod mod ( ( ( ]/ [ ]/ [ > p q q g pf h q f p f p q q p g f h q p Z Z Z (mod ]/ [ ]/ [ g pf h g f q p Z Z

27 NTRU (3 (mod ]/ [ ]/ [ ]/ [ q q p q rh m e r e r m Z Z Z (mod rh m e r m

28 (mod (mod (mod p f c p a q fe a NTRU (3 (mod (mod (mod ( ( p m fm f f c p fm prg fm a q prg fm g rpf m f rh m f fe a (mod 3 (mod f c a fe a e

29 NTRU (4 CoppersmithShamir(EC97 (mod ( q g pf h q p ' ' g f (mod q y h p λ y q q q h h h h h h h h h qi H I B M O M M M O M M M O M M M O M M λ λ λ λ / ( (mod g f q h p h i λ

30 Howgrave-Graham (997 < N /d Coppersmith dual lattice(b -t shrik p ( a 4 3 p( c d e f N N N N N N X N NXa NX 3 NX NX a X 3 4 f Xe X d X c X Xf X e X d X c X p ( p( p ( p( X X X 3 X 4 X 5 X r(

31 Howgrave-Graham p ( 4 9 (mod 35 4 ( p ( X ( r(

32 k(as (mod e d < N.9 RSA ed (mod (p- q- (p-(q-/(n-p-q/ edk{(n/ - (pq/} A (N/ s (pq/ dn k s e N s e.5 k e Wieer IT9 <.5 Boeh C99 Howgrave-Graham <.9 D.Boeh G. Durfee Cryptaalysis of RSA with Private Key d ess tha N.9 CRYPTO 99

33 Np r q p P tp-p (Pt r (mod p r f((p r t f( (mod p r p r p r N f( (mod N Howgrave-Graham P r > (log p /

34 Hidde suset sum prolem (C99 ai i C Noisy polyomial reductio (Noisy chiise remaider theorem (E NFS Orthogoal lattice techique Qu-Vastoe Nguye C97 Ajtai-Dwork Nguye C98 SAC97 Nguye SAC98 Hidde suset sum Nguye C99 GGH Goldreich C97 Nguye C99

35 Jaco Buhler 97 NFS oud : /3 two-third method Vallee Math.Comp.??

36 (C GMP

37 mappig Hastad Coppersmith estra-estra-ovasz Schorr Korkie-Zolotarev

38 (B (UB U G - d( (B (B d(b d(b B B' U B UB (B (B det(b 8996 det(b det(u -

39 B ? B' d(b /4 37.

40 4 O ( log B B ( ma{ } B ij

41 µ ij j < i * * 3 * i µ ii i 4 i < i * d( ( / 4 / d( {... } j t... t j ma{... t } for j...t

42 ( a ( a a K a N αamod N ( αa mod N αa mod N K αa mod N α B ε a a a N N M M M O M N ε a NI ( B (B ( (^^;

43 ( α k k K k Z (B ( ε a a a α ( N k ( N k M M ( N k ( αε αa k N αa k N K αa k N B ( ( αε αa kn αa kn K αa kn ( αε αa mod N αa mod N K αa mod N B B B d( B ( N ε / 4 /( / 4 /(

44 a ( N9 a mod N ( a 77.8 B B' ( ( ( ( ( ( mod N 33 mod N 58 mod N 3 mod N (

45 ( a ( a a K a C N a a a C(mod ( K B a a M M O M M M a C N I t a C N ( B (B

46 ( K t k Z (B ( a ( a ( M a ( C ( N ( K t a a tc kn ( K t( a a tcmod N M t k B ( - d( B ( / 4 /( ( / 4 /( N

47 (mod B B' ( - ( t t ( :345*(-3333*(-358*5 33 (mod d(b / 543 /5 6.88

48 ( ( λa λa B M λa ( B ( λx B δ

49 (mod 543 B 345λ 3333λ 58λ λ 4996λ 543λ 3 ( B' B' :345*63333* (mod 543

50 a a a c(mod N a ( a a c mod N 3 3 B β N βa βa βc a 3 t 3 mod N Z ( B ( β( a a tcmod N t

51 Hastad (988 d p( (mod N <N /d(d (p( p d d p d- d- p p p B < X d d ε X p X p Xp p d d d X N d X N M M M O M M XN N p ( (mod p'(

52 Hastad p ( (mod 399 ε X B X 33X 85 X N XN N 8 8X 7X X 7X X 95X 5 N

53 Coppersmith (996 d p( (mod N <N /d < X p X p M M O M M M B ( d X pd d X p d N d B p( i pz i i (mod zi X i p( p(... p( p( p(... z i p( 3 p( 3... ( (... d- d

54 Coppersmith B X t t t t p p p p N N N N p p p p f X a e f X a d e 3 X c d 4 X c 5 X N N N N p ( a 4 3 p( c d e f ( (

55 ( (!!

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