ACM Ubiquity, Volume 9, Issue 20 May 20 26, 2008 U s i n g R S A , - PDF document

ACM Ubiquity, Volume 9, Issue 20 May 20 – 26, 2008 E lli p t i c C u r v e C r y p t og r a ph y Vivek Kapoor Vivek Sonny Abraham Department of Computer Engineering Department of Computer Engineering Delhi College of Engineering Delhi College of Engineering Ramesh Singh National Informatics Centre Government Of India A b s t r a c t Proof of Delivery T h e s e nd e r s h o u l d b e a b le t o p r o v e t h a t t h e r eci p ie n t r ecei v e d t h e m e ss a ge . T h i s p a p e r d e s c r i b e s t h e E lli p t ic C u r v e C r y p t og r a ph y a lgo r i t h m a nd i t s s u i t a b ili t y f o r 2 H i s t o r y s m a r t c a r d s . C r y p t og r a ph y h a s b ee n i n u s e f o r ce n t u r ie s n ow , a nd 1 I n f o r m a t i o n S e c u r i t y t h e e a r lie s t ci ph e r s we r e ei t h e r u s e d t r a n s p o s i t io n o r s ub s t i t u t io n, a nd m e ss a ge s we r e e n co d e d a nd d e- I n f o r m a t io n s ec u r i t y i s e ss e n t i a l f o r t o d a y ’ s wo r l d co d e d b y h a nd. H owe v e r , t h e s e s c h e m e s s a t i s fie d o n l y s i n ce , f o r p r ofi t a b le a nd leg a l t r a d i n g , co n fi d e n t i a l- t h e b a s ic r e q u i r e m e n t o f co n fi d e n t i a li t y . I n m o r e r e- i t y , i n t eg r i t y a nd n o n - r e pud i a b ili t y o f t h e a ss oci a t e d ce n t t i m e s , wi t h t h e i n v e n t io n o f p r oce ss i n g m a c h i n e s , i n f o r m a t io n a r e n ece ss a r y . T h i s c a n b e d o n e u s i n g m o r e r o bu s t a lgo r i t h m s we r e r e q u i r e d, a s t h e s i m p le c r y p t og r a ph ic s y s t e m s . I n t ge r a t e d c r y p t og r a ph ic s y s - ci ph e r s we r e e a s y t o d eco d e u s i n g t h e s e m a c h i n e s , t e m s s a t i s f y a ll t h e a b o v e- m e n t io n e d r e q u i r e m e n t s . a nd m o r eo v e r t h e y d i d n o t h a v e a n y o f t h e a f o r e D e s i r e d p r o p e r t ie s o f a s ec u r e co mm un ic a t io n s y s t e m m e n t io n e d p r o p e r t ie s . Sec u r e d a t a co mm un ic a t io n b ec a m e a n ece ss i t y i n t h e 20 t h ce n t u r y a nd a lo t o f m a y i n cl ud e a n y o r a ll o f t h e f ollowi n g [ wi k , P V O 96 ] : r e s e a r c h w a s d o n e i n t h i s fiel d b y go v e r n m e n t a ge n - Confidentiality O n l y a n a u t h o r ize d r eci p ie n t cie s , du r i n g a nd f ollowi n g t h e wo r l d -w a r s . T h e m o s t s h o u l d b e a b le t o e x t r a c t t h e co n t e n t s o f t h e f a m o u s m a c h i n e o f t h i s t i m e , Enigma w a s a n elec t r o- e n co d e d d a t a , i n p a r t o r w h ole . m ec h a n ic a l d e v ice w h ic h w a s u s e d b y t h e G e r m a n A r m y . Integrity T h e r eci p ie n t s h o u l d b e a b le t o e s t a b li s h i f t h e m e ss a ge h a s b ee n a l t e r e d du r i n g t r a n s m i s - s io n. 2 . 1 S y mm e t r i c A l go r i t h m s Authentication T h e r eci p ie n t s h o u l d b e a b le t o T h e fi r s t s ec r e t k e y - b a s e d c r y p t og r a ph ic a lgo r i t h m s i d e n t i f y t h e s e nd e r , a nd v e r i f y t h a t t h e pu r - wo r k e d o n t h e s y mm e t r ic a lgo r i t h m s . T h e y a ss u m e d p o r t e d s e nd e r a c t u a ll y s e n t t h e m e ss a ge . t h a t b o t h co mm un ic a t i n g p a r t ie s s h a r e d s o m e s ec r e t i n f o r m a t io n, w h ic h w a s un i q u e t o t h e m , m u c h li k e Non-Repudiation T h e s e nd e r s h o u l d n o t b e a b le t h e ol d e r One Time Pads . U s i n g t h i s s ec r e t i n f o r - t o d e n y s e nd i n g t h e m e ss a ge , i f h e a c t u a ll y d i d m a t io n, a l s o c a lle d a key , t h e s e nd e r e n c r y p t e d 1 t h e s e nd i t . d a t a , a nd t h e r eci p ie n t w a s a b le t o d ec r y p t . S upp o s e Alice w a n t s t o s e nd a m e ss a ge m t o Bob , a nd a ss u m e Anti-replay T h e m e ss a ge s h o u l d n o t b e a llowe d t o b e s e n t t o m u l t i m p le r eci p ie n t s , wi t h o u t t h e 1 encrypt-encipher-encode and decrypt-decipher-decode are s e nd e r ’ s k n owle d ge . used interchangeably 1

ACM Ubiquity, Volume 9, Issue 20 May 20 – 26, 2008 t h a t t h e y b o t h h a v e a l r e a d y s h a r e d a k e y k . Alice 1 . Alice p a ss e s t h e m e ss a ge m a nd Bob’s pub lic k e y B ∗∗ t o a n a pp r o p r i a t e e n c r y p t io n a lgo r i t h m t o e n c r y p t s m u s i n g t h e s h a r e d k e y k t o ge t t h e ci ph e r t e x t . co n s t r u c t t h e e n c r y p t e d m e ss a ge . C (Σ B ,m ) = E Σ B ( m ) ( 3 ) C ∗ ( k,m ) = E † k ( m ) ( 1 ) Bob c a n t h e n d ec r y p t t h i s m e ss a ge u s i n g h i s co p y o f 2 . Alice t r a n s m i t s t h e e n co d e d m e ss a ge t o Bob . t h e k e y k , a nd e x t r a c t t h e o r igi n a l m e ss a ge m . 3 . Bob d ec r y p t s t h e e n c r y p t e d m e ss a ge r ecei v e d b y D ‡ k ( C ( k,m ) ) = D k ( E k ( m )) = m ( 2 ) h i m , u s i n g h i s p r i v a t e k e y ∆ B § a nd t h e a pp r o- p r i a t e d ec r y p t io n a lgo r i t h m . T h i s t ec hn i q u e t h o u g h s i m p le a nd e a s y t o i m p le m e n t , h a s o b v io u s d r a w b a c k s , s o m e o f w h ic h a r e li s t e d h e r e: D ∆ B ( C (Σ B ,m ) ) = D ∆ B ( E Σ B ( m )) = m ( 4 ) • A s h a r e d s ec r e t k e y m u s t b e a g r ee d up o n b y b o t h p a r t ie s . Bob i s a ss u r e d t h a t t h e d a t a h e r ecie v e d i s n o t t a m - p e r e d wi t h o r le a k e d, a s o n l y h i s p r i v a t e k e y c a n • I f a u s e r h a s n co mm un ic a t i n g p a r t n e r s , t h e n d ec r y p t t h e d a t a . Si m il a r l y Bob c a n s e nd d a t a t o n s ec r e t k e y s m u s t b e m a i n t a i n e d, o n e f o r e a c h Alice u s i n g h e r pub lic k e y A . T h e P K C s c h e m e p a r t n e r . a l s o s a t i s fie s t h e N o n - R e pud i a t io n a nd A u t h e n t ic- i t y b y u s i n g i nn o v a t i v e t ec hn i q u e s s u c h a s D igi t a l • A u t h e n t ici t y o f o r igi n o r r ecei p t c a nn o t b e Sig n a t u r e s [ Sc h 95 ] . p r o v e d b ec a u s e t h e s ec r e t k e y i s s h a r e d. • M a n a ge m e n t o f t h e s y mm e t r ic k e y s b eco m e s 3 S m a r t C a r d s p r o b le m a t ic . 3 . 1 B a s i c s 2 . 2 P ub li c K e y c r y p t og r a ph y A s m a r t c a r d, c h i p c a r d, o r i n t eg r a t e d ci r c u i t ( s ) T h e co n ce p t o f P ub lic K e y c r y p t og r a ph y ( P K C ) w a s c a r d ( I CC ) , i s d efi n e d a s a n y p oc k e t - s ize d c a r d wi t h fi r s t i n t r o du ce d b y D i e a nd H ell m a n i n 1976 , i n e m b e dd e d i n t eg r a t e d ci r c u i t s . A l t h o u g h t h e r e i s a t h ei r s e m i n a l p a p e r , N ew D i r ec t io n s i n C r y p t og r a ph y d i v e r s e r a n ge o f a pp lic a t io n s , t h e r e a r e t wo b r o a d [ D H 76 ] . T h i s p a p e r a l s o a dd r e ss e d t h e i ss u e o f k e y c a t ego r ie s o f I CC s . M e m o r y c a r d s co n t a i n o n l y e x c h a n ge , b a s e d o n t h e i n t r a c t a b ili t y o f t h e d i s c r e t e n o n - v ol a t ile m e m o r y s t o r a ge co m p o n e n t s , a nd p e r - log a r i t h m p r o b le m . I n a pub lic k e y c r y p t o s y s t e m , h a p s s o m e s p ecific s ec u r i t y logic . M ic r o p r oce ss o r e a c h u s e r h a s a p a i r o f k e y s , o n e pub li s h e d pub licl y , c a r d s co n t a i n m e m o r y a nd m ic r o p r oce ss o r co m p o- k n ow n a s t h e public key , a nd t h e o t h e r k n ow n a s a n e n t s . T h e s t a nd a r d p e r ce p t io n o f a smart card private key , i s s t o r e d i n a s ec u r e loc a t io n. P ub lic i s a m ic r o p r oce ss o r c a r d o f c r e d i t -c a r d d i m e n s io n s k e y c r y p t o s y s t e m s r el y o n t h e e x i s t e n ce o f a t r a p - ( o r s m a lle r , e . g . t h e G S M S I M c a r d ) wi t h v a r io u s d oo r f un c t io n, w h ic h m a k e s d eco d i n g p o ss i b le gi v e n t a m p e r - r e s i s t a n t p r o p e r t ie s ( e . g . a s ec u r e c r y p t o- t h e k n owle d ge o f t h e p r i v a t e k e y co rr e s p o nd i n g t o t h e p r oce ss o r , s ec u r e file s y s t e m , hu m a n - r e a d a b le f e a - pub lic k e y f o r e n c r y p t io n. C o n s i d e r i n g a c a s e a n a l- t u r e s ) a nd i s c a p a b le o f p r o v i d i n g s ec u r i t y s e r v ice s ogo u s t o t h e o n e d e s c r i b e d i n t h e c a s e o f s y mm e t r ic ( e . g . co n fi d e n t i a li t y o f i n f o r m a t io n i n t h e m e m o r y ) . k e y s , w h e r e b y Alice wi s h e s t o s e nd a m e ss a ge m t o N o t a ll c h i p c a r d s co n t a i n a m ic r o p r oce ss o r ( eg . t h e Bob . t h e f ollowi n g s t e p s will a cco m p li s h t h e t a s k : m e m o r y c a r d s ) , t h e r e f o r e n o t a ll c h i p c a r d s a r e n ec- ∗ represents the cipher text corresponding to message m and e ss a r il y a l s o s m a r t c a r d s [ wi k ] . key k † represents the Encryption function ∗∗ Σ χ represents the published public key of user χ ‡ represents the Decryption function § ∆ τ represents the secure private key of user τ 2