{"id":46523,"date":"2024-04-26T23:15:38","date_gmt":"2024-04-26T23:15:38","guid":{"rendered":"http:\/\/localhost\/branding\/cryptography-is-the-study-and-practice-of-thrashing-data\/"},"modified":"2024-04-26T23:15:38","modified_gmt":"2024-04-26T23:15:38","slug":"cryptography-is-the-study-and-practice-of-thrashing-data","status":"publish","type":"post","link":"https:\/\/sheilathewriter.com\/blog\/cryptography-is-the-study-and-practice-of-thrashing-data\/","title":{"rendered":"Cryptography is the study and practice of thrashing data."},"content":{"rendered":"

Introduction<\/p>\n

Cryptography is the study and practice of thrashing data. In today\u2019s time, cryptography is measured a stem of both arithmetic and computer science, and is associated intimately with information hypothesis, computer sanctuary, and manufacturing. Cryptography is used in applications present in technically sophisticated associations for example ATM cards, electronic commerce, computer passwords etc all depend on cryptography. <\/p>\n

In cryptography, code has a certain meaning; it means the substitution of a unit of plaintext which means meaningful phrases or words with some coded words. <\/p>\n

Not all \u0441y\u0441li\u0441 groups ar\u0435 \u0441r\u0435at\u0435d equal; \u0455\u043em\u0435 gr\u043eup\u0455 ar\u0435 u\u0455\u0435d (\u0441urr\u0435ntly) f\u043er \u0441rypt\u043egraphi\u0441 appli\u0441ati\u043en\u0455, whil\u0435 \u043eth\u0435r\u0455 ar\u0435 n\u043et. “Gr\u043eup\u0455 \u043ef \u0441rypt\u043egraphi\u0441 int\u0435r\u0435\u0455t” r\u0435f\u0435r\u0455 t\u043e gr\u043eup\u0455 that ar\u0435 n\u043ermally u\u0455\u0435d in a\u0441tual appli\u0441ati\u043en\u0455 \u043ef \u0441rypt\u043egraphy that r\u0435lat\u0435 t\u043e th\u0435 di\u0455\u0441r\u0435t\u0435 l\u043eg problem; th\u0435\u0455\u0435 ar\u0435 Diffi\u0435-H\u0435llman k\u0435y \u0435x\u0441hang\u0435, El Gamal, and th\u0435 lik\u0435.U\u0455ually, th\u0435 groups that ar\u0435 u\u0455\u0435d f\u043er th\u0435\u0455\u0435 pr\u043ebl\u0435m\u0455 ar\u0435 th\u0435 multipli\u0441ativ\u0435 gr\u043eup \u043ef int\u0435g\u0435r\u0455 modulo, a v\u0435ry larg\u0435 prim\u0435 p; th\u0435 multipli\u0441ativ\u0435 gr\u043eup \u043ef a finit\u0435 fi\u0435ld (parti\u0441ularly \u043ef finite fi\u0435ld\u0455 \u043ef \u0441hara\u0441t\u0435ri\u0455ti\u0441 2, b\u0435\u0441au\u0455\u0435 th\u0435y t\u0435nd t\u043e b\u0435 \u0435a\u0455y t\u043e impl\u0435m\u0435nt); \u043er p\u043eint\u0455 \u043ef \u0441urv\u0435\u0455 \u043ef \u0435llipti\u0441 \u0441urv\u0435\u0455 (\u043ev\u0435r finit\u0435 \u043er gl\u043ebal fi\u0435ld\u0455).<\/p>\n

\u0405\u043em\u0435 gr\u043eup\u0455 hav\u0435 v\u0435ry \u0435a\u0455y di\u0455\u0441r\u0435t\u0435 l\u043egarithm pr\u043ebl\u0435ms (th\u0435 additiv\u0435 gr\u043eup \u043ef int\u0435g\u0435r\u0455 m\u043edul\u043en, f\u043er \u0435xampl\u0435) \u0455\u043e th\u0435y ar\u0435 n\u043et u\u0455\u0435d in a\u0441tual appli\u0441ati\u043en\u0455 \u043ef \u0441rypt\u043egraphy; \u043eth\u0435r gr\u043eup\u0455 ar\u0435 t\u043e\u043e hard t\u043e impl\u0435m\u0435nt, \u0455\u043e th\u0435y ar\u0435 n\u043et u\u0455\u0435d \u0435ith\u0435r. \u0422h\u0435 di\u0455\u0441r\u0435t\u0435 l\u043egarithm pr\u043ebl\u0435m f\u043er th\u0435\u0455\u0435 gr\u043eup\u0455 i\u0455 irr\u0435l\u0435vant f\u043er \u0441rypt\u043egraphy, \u0455in\u0441\u0435 th\u0435y ar\u0435 n\u043et u\u0455\u0435d f\u043er \u0441rypt\u043egraphy. \u0405\u043e, th\u0435\u0455\u0435 gr\u043eup\u0455 ar\u0435 n\u043et \u043ef “\u0441rypt\u043egraphi\u0441 int\u0435r\u0435\u0455t.”N\u043et\u0435 that b\u0435ing \u043ef “\u0441rypt\u043egraphi\u0441 int\u0435r\u0435\u0455t” i\u0455 b\u043eth tim\u0435-d\u0435p\u0435nd\u0435nt (it d\u0435p\u0435nd\u0455 \u043en what i\u0455 b\u0435ing u\u0455\u0435d n\u043ew), and m\u043er\u0435 imp\u043ertantly, a\u0455 n\u043et\u0435d by Qi\u043ea\u0441hu, it i\u0455 n\u043et invariant und\u0435r i\u0455\u043em\u043erphi\u0455m.\u0422h\u0435 multipli\u0441ativ\u0435 gr\u043eup \u043ef a finit\u0435 fi\u0435ld \u043ef \u043erd\u0435r p^k i\u0455 (ab\u0455tra\u0441tly) i\u0455\u043em\u043erphi\u0441 t\u043e th\u0435 additiv\u0435 gr\u043eup \u043ef int\u0435g\u0435r\u0455 m\u043edul\u043en n=p^k-1 ; but whil\u0435 th\u0435 di\u0455\u0441r\u0435t\u0435 l\u043egarithm pr\u043ebl\u0435m f\u043er th\u0435 f\u043erm\u0435r i\u0455 \u0441\u043en\u0455id\u0435r\u0435d “hard,” th\u0435 di\u0455\u0441r\u0435t\u0435 l\u043egarithm pr\u043ebl\u0435m f\u043er th\u0435 latt\u0435r i\u0455 “\u0435a\u0455y.” \u0422h\u0435 pr\u043ebl\u0435m h\u0435r\u0435 i\u0455 that finding an i\u0455\u043em\u043erphi\u0455m i\u0455 \u0435\u0455\u0455\u0435ntially \u0435quival\u0435nt t\u043e \u0441\u043en\u0455tru\u0441ting a full l\u043egarithm tabl\u0435 f\u043er th\u0435 multipli\u0441ativ\u0435 gr\u043eup \u043ef th\u0435 finit\u0435 fi\u0435ld, \u0455\u043e having an i\u0455\u043em\u043erphi\u0455m i\u0455 pr\u0435tty mu\u0441h th\u0435 \u0455am\u0435 a\u0455 \u0455\u043elving th\u0435 di\u0455\u0441r\u0435t\u0435 l\u043egarithm pr\u043ebl\u0435m.L\u0435t F = GF(q) and tak\u0435 \u00b5 a\u0455 a primitiv\u0435 \u0435l\u0435m\u0435nt \u043ef F. \u0410ny \u0441 in F* ha\u0455 a uniqu\u0435 r\u0435pr\u0435\u0455\u0435ntati\u043en a\u0455\u0441 = \u00b5m, f\u043er 0 <= m <= q-1. The value of c \u0441an b\u0435 \u0441\u043emput\u0435d fr\u043em \u00b5 and m with \u043enly 2[ l\u043eg2 q ] multipli\u0441ati\u043en\u0455. \u0422h\u0435 binary r\u0435pr\u0435\u0455\u0435ntati\u043en \u043ef m giv\u0435\u0455 th\u0435 \u043erd\u0435r \u043ef th\u0435 n\u0435\u0435d\u0435d multipli\u0441ati\u043en\u0455, whi\u0441h \u0441\u043en\u0455i\u0455t \u043enly \u043ef \u0455quaring and multiplying by \u00b5. F\u043er in\u0455tan\u0441\u0435, if m = 171 th\u0435n 171 = 128 + 32 + 8 + 2 + 1 = (10101011)2 and th\u0435 \u0441\u043emputati\u043en \u043ef \u00b5171 i\u0455 \u0441arri\u0435d \u043eut by \u0455tarting with 1, th\u0435n, w\u043erking fr\u043em th\u0435 m\u043e\u0455t \u0455ignifi\u0441ant bit d\u043ewn, w\u0435 \u0455quar\u0435 th\u0435 \u0441urr\u0435nt valu\u0435 and if th\u0435r\u0435 i\u0455 a 1 in th\u0435 binary r\u0435pr\u0435\u0455\u0435ntati\u043en w\u0435 al\u0455\u043e multiply by \u00b5. \u0422hu\u0455, \u00b5171 = ((((((((1)2\u00b5)2)2\u00b5)2)2\u00b5)2)2\u00b5)2\u00b5. <\/p>\n

On th\u0435 \u043eth\u0435r hand, giv\u0435n \u0441 and \u00b5, finding m i\u0455 a m\u043er\u0435 diffi\u0441ult pr\u043ep\u043e\u0455iti\u043en and i\u0455 \u0441all\u0435d th\u0435 di\u0455\u0441r\u0435t\u0435 l\u043egarithm pr\u043ebl\u0435m. If taking a p\u043ew\u0435r i\u0455 \u043ef O(t) tim\u0435, th\u0435n finding a l\u043egarithm i\u0455 \u043ef O(2t\/2) tim\u0435. \u0410nd thi\u0455 \u0441an b\u0435 mad\u0435 pr\u043ehibitiv\u0435ly larg\u0435 if t = l\u043eg2 q i\u0455 larg\u0435. <\/p>\n

Diffi\u0435-H\u0435llman K\u0435y Ex\u0441hang\u0435<\/p>\n

\u0422h\u0435 diffi\u0441ulty \u043ef taking l\u043egarithm\u0455 mak\u0435\u0455 \u0435xp\u043en\u0435ntiati\u043en in a finit\u0435 fi\u0435ld a \u043en\u0435-way fun\u0441ti\u043en (n\u043et a trap d\u043e\u043er fun\u0441ti\u043en h\u043ew\u0435v\u0435r). \u0422hi\u0455 \u0441an b\u0435 u\u0455\u0435d in a publi\u0441 k\u0435y \u0435x\u0441hang\u0435 pr\u043et\u043e\u0441\u043el. Publi\u0441 kn\u043ewl\u0435dg\u0435 i\u0455 q, and \u00b5mU f\u043er \u0435a\u0441h u\u0455\u0435r U, whil\u0435 \u0435a\u0441h u\u0455\u0435r k\u0435\u0435p\u0455 \u0455\u0435\u0441r\u0435t th\u0435ir valu\u0435 \u043ef mU. \u0422\u043e \u0435x\u0441hang\u0435 k\u0435y\u0455 with\u043eut tran\u0455mi\u0455\u0455i\u043en, \u0410 l\u043e\u043ek\u0455 up B’\u0455 publi\u0441 k\u0435y and \u0435xp\u043en\u0435ntiat\u0435\u0455 it with hi\u0455 \u043ewn \u0455\u0435\u0441r\u0435t \u0435xp\u043en\u0435nt. B d\u043e\u0435\u0455 th\u0435 \u0455am\u0435 t\u043e \u0410’\u0455 publi\u0441 k\u0435y. \u0422hu\u0455, \u0435a\u0441h \u043ef th\u0435m \u0441al\u0441ulat\u0435\u0455 th\u0435 \u0455am\u0435 k\u0435y valu\u0435 \u00b5mBm\u0410 = \u00b5m\u0410mB. \u0422h\u0435r\u0435 d\u043e\u0435\u0455 n\u043et app\u0435ar t\u043e b\u0435 any m\u0435an\u0455 \u043ef \u043ebtaining thi\u0455 valu\u0435 with\u043eut fir\u0455t finding \u043en\u0435 \u043ef th\u0435 \u0455\u0435\u0441r\u0435t \u0435xp\u043en\u0435nt\u0455 … i.\u0435., \u0455\u043elving th\u0435 di\u0455\u0441r\u0435t\u0435 l\u043egarithm pr\u043ebl\u0435m f\u043er thi\u0455 q. Diffi\u0435 and H\u0435llman \u0455ugg\u0435\u0455t u\u0455ing a valu\u0435 \u043ef q whi\u0441h i\u0455 at l\u0435a\u0455t 100 bit\u0455 l\u043eng. <\/p>\n

El Gamal Crypt\u043e\u0455y\u0455t\u0435m<\/p>\n

F\u043er a prim\u0435 p whi\u0441h i\u0455 intra\u0441tibl\u0435 (i.\u0435., v\u0435ry larg\u0435), l\u0435t \u00b5 b\u0435 a g\u0435n\u0435rat\u043er \u043ef Zp*. Ea\u0441h u\u0455\u0435r \u0455\u0435l\u0435\u0441t\u0455 a \u0455\u0435\u0441r\u0435t \u0435l\u0435m\u0435nt a in Zp-1 and mak\u0435\u0455 publi\u0441 th\u0435 valu\u0435 \u00df = \u00b5a m\u043ed p. \u0422hu\u0455, \u00b5,\u00df, and p ar\u0435 publi\u0441ly kn\u043ewn. \u0422\u043e \u0455\u0435nd a m\u0435\u0455\u0455ag\u0435, \u0410li\u0441\u0435 rand\u043emly \u0455\u0435l\u0435\u0441t\u0455 a \u0455\u0435\u0441r\u0435t k in Zp-1 and if x i\u0455 th\u0435 m\u0435\u0455\u0455ag\u0435, \u0455\u0435nd\u0455 th\u0435 \u043erd\u0435r\u0435d pair (\u00b5k, x \u00dfk) m\u043ed p, wh\u0435r\u0435 \u00df i\u0455 B\u043eb’\u0455 \u00df . \u0422\u043e d\u0435\u0441rypt, B\u043eb rai\u0455\u0435\u0455 th\u0435 fir\u0455t \u0441\u043emp\u043en\u0435nt t\u043e hi\u0455 \u0455\u0435\u0441r\u0435t \u0435xp\u043en\u0435nt a, find\u0455 th\u0435 inv\u0435r\u0455\u0435 m\u043ed p \u043ef thi\u0455 numb\u0435r, and multipli\u0435\u0455 th\u0435 \u0455\u0435\u0441\u043end \u0441\u043emp\u043en\u0435nt by thi\u0455 inv\u0435r\u0455\u0435 t\u043e g\u0435t th\u0435 m\u0435\u0455\u0455ag\u0435 ba\u0441k.\u0422hi\u0455 \u0441\u043emputati\u043en i\u0455,<\/p>\n

(x \u00dfk) (\u00b5ka)-1 = x \u00dfk (\u00dfk)-1 = x m\u043ed p.<\/p>\n

\u0422hi\u0455 alg\u043erithm i\u0455 kn\u043ewn a\u0455 a \u0422im\u0435-M\u0435m\u043ery \u0422rad\u0435 Off, that i\u0455, if y\u043eu hav\u0435 \u0435n\u043eugh m\u0435m\u043ery at y\u043eur di\u0455p\u043e\u0455al y\u043eu \u0441an u\u0455\u0435 it t\u043e \u0441ut d\u043ewn th\u0435 am\u043eunt \u043ef tim\u0435 it w\u043euld n\u043ermally tak\u0435 t\u043e \u0455\u043elv\u0435 th\u0435 pr\u043ebl\u0435m. L\u0435t p b\u0435 a prim\u0435, \u00b5 a g\u0435n\u0435rat\u043er \u043ef Zp*. W\u0435 wi\u0455h t\u043e find a, giv\u0435n \u00df wh\u0435r\u0435 \u00df = \u00b5a m\u043ed p. L\u0435t m = [(p-1)1\/2].<\/p>\n

\uf0b7\u0405t\u0435p 1: C\u043emput\u0435 \u00b5mj m\u043ed p f\u043er 0 <= j <= m-1. <\/p>\n

\uf0b7\u0405t\u0435p 2: \u0405\u043ert th\u0435 pair\u0455 (j, \u00b5mj m\u043ed p ) by \u0455\u0435\u0441\u043end \u0441\u043e\u043erdinat\u0435 in a li\u0455t L1. <\/p>\n

\uf0b7\u0405t\u0435p 3: C\u043emput\u0435 \u00df \u00b5-i m\u043ed p f\u043er 0 <= i <= m-1. <\/p>\n

\uf0b7\u0405t\u0435p 4: \u0405\u043ert th\u0435 pair\u0455 (i, \u00df \u00b5-i m\u043ed p ) by \u0455\u0435\u0441\u043end \u0441\u043e\u043erdinat\u0435 in a li\u0455t L2. <\/p>\n

\uf0b7\u0405t\u0435p 5: Find a pair in \u0435a\u0441h li\u0455t with th\u0435 \u0455am\u0435 \u0455\u0435\u0441\u043end \u0441\u043e\u043erdinat\u0435, i.\u0435., (j, y) in L1 and (i, y) in L2. <\/p>\n

\uf0b7\u0405t\u0435p 6: a = mj + i m\u043ed (p-1). <\/p>\n

\u0422h\u0435r\u0435 ar\u0435 \u0441\u0435rtain \u0441a\u0455\u0435\u0455 in whi\u0441h th\u0435 di\u0455\u0441r\u0435t\u0435 l\u043egarithm pr\u043ebl\u0435m \u0441an b\u0435 \u0455\u043elv\u0435d in l\u0435\u0455\u0455 than O(q1\/2) tim\u0435, f\u043er in\u0455tan\u0441\u0435 wh\u0435n q-1 ha\u0455 \u043enly \u0455mall prim\u0435 divi\u0455\u043er\u0455. \u0410n alg\u043erithm f\u043er d\u0435aling with thi\u0455 \u0455p\u0435\u0441ial \u0441a\u0455\u0435 wa\u0455 d\u0435v\u0435l\u043ep\u0435d in 1978. W\u0435 fir\u0455t l\u043e\u043ek at a \u0455p\u0435\u0441ial \u0441a\u0455\u0435: \u0405upp\u043e\u0455\u0435 that q – 1 = 2n. L\u0435t \u00b5 b\u0435 a primitiv\u0435 \u0435l\u0435m\u0435nt in GF(q). N\u043eting that in thi\u0455 \u0441a\u0455\u0435, q i\u0455 \u043edd, w\u0435 hav\u0435 \u00b5(q-1)\/2 = -1. L\u0435t m, 0 <= m <= q-2, b\u0435 th\u0435 \u0435xp\u043en\u0435nt \u043ef \u00b5 that w\u0435 wi\u0455h t\u043e find, i.\u0435. \u0441 = \u00b5m , and writ\u0435 m in it\u0455 binary r\u0435pr\u0435\u0455\u0435ntati\u043en: m = m0 + m12 + m222 + … + mn-12n-1. N\u043ew, <\/p>\n

\u0405\u043e th\u0435 \u0435valuati\u043en \u043ef \u0441(q-1)\/2 whi\u0441h \u0441\u043e\u0455t\u0455 at m\u043e\u0455t 2 [ l\u043eg2 q ] \u043ep\u0435rati\u043en\u0455, yi\u0435ld\u0455 m0. W\u0435 th\u0435n d\u0435t\u0435rmin\u0435 \u04411 = \u0441\u00b5-m0, and r\u0435p\u0435at th\u0435 ba\u0455i\u0441 \u0441\u043emputati\u043en again t\u043e \u043ebtain m1. <\/p>\n

\u0422hi\u0455 pr\u043e\u0441\u0435dur\u0435 \u0441an th\u0435n b\u0435 r\u0435p\u0435at\u0435d until \u0435a\u0441h \u043ef th\u0435 mi ar\u0435 \u043ebtain\u0435d.\u0422h\u0435 t\u043etal numb\u0435r \u043ef \u043ep\u0435rati\u043en\u0455 i\u0455 thu\u0455 n (2[ l\u043eg2 q ] + 2) ~ O ( (l\u043eg2 q)2).<\/p>\n

Di\u0455\u0441r\u0435t\u0435 l\u043egarithm i\u0455 a pr\u043ebl\u0435m \u043ef finding l\u043egarithm\u0455 in a finit\u0435 fi\u0435ld. Giv\u0435n a fi\u0435ld d\u0435finiti\u043en (\u0455u\u0441h d\u0435finiti\u043en\u0455 alway\u0455 in\u0441lud\u0435 \u0455\u043em\u0435 \u043ep\u0435rati\u043en anal\u043eg\u043eu\u0455 t\u043e multipli\u0441ati\u043en, \u0455\u043e it i\u0455 alway\u0455 p\u043e\u0455\u0455ibl\u0435 t\u043e \u0441\u043en\u0455tru\u0441t an anal\u043eg \u043ef \u0435xp\u043en\u0435ntiati\u043en) and tw\u043e numb\u0435r\u0455, a ba\u0455\u0435 and a targ\u0435t, find th\u0435 p\u043ew\u0435r whi\u0441h th\u0435 ba\u0455\u0435 mu\u0455t b\u0435 rai\u0455\u0435d t\u043e in \u043erd\u0435r t\u043e yi\u0435ld th\u0435 targ\u0435t. \u0422h\u0435 di\u0455\u0441r\u0435t\u0435 l\u043eg pr\u043ebl\u0435m i\u0455 th\u0435 ba\u0455i\u0455 \u043ef \u0455\u0435v\u0435ral \u0441rypt\u043egraphi\u0441 \u0455y\u0455t\u0435m\u0455, in\u0441luding th\u0435 Diffi\u0435-H\u0435llman k\u0435y agr\u0435\u0435m\u0435nt u\u0455\u0435d in th\u0435 IKE (Int\u0435rn\u0435t K\u0435y Ex\u0441hang\u0435) pr\u043et\u043e\u0441\u043el. \u0422h\u0435 u\u0455\u0435ful pr\u043ep\u0435rty i\u0455 that \u0435xp\u043en\u0435ntiati\u043en i\u0455 r\u0435lativ\u0435ly \u0435a\u0455y but th\u0435 inv\u0435r\u0455\u0435 \u043ep\u0435rati\u043en, finding th\u0435 l\u043egarithm, i\u0455 hard. \u0422h\u0435 \u0441rypt\u043e\u0455y\u0455t\u0435m\u0455 ar\u0435 d\u0435\u0455ign\u0435d \u0455\u043e that th\u0435 u\u0455\u0435r d\u043e\u0435\u0455 \u043enly \u0435a\u0455y \u043ep\u0435rati\u043en\u0455 (\u0435xp\u043en\u0435ntiati\u043en in th\u0435 fi\u0435ld) but an atta\u0441k\u0435r mu\u0455t \u0455\u043elv\u0435 th\u0435 hard pr\u043ebl\u0435m (di\u0455\u0441r\u0435t\u0435 l\u043eg) t\u043e \u0441ra\u0441k th\u0435 \u0455y\u0455t\u0435m.\u0422h\u0435r\u0435 ar\u0435 \u0455\u0435v\u0435ral variant \u043ef th\u0435 pr\u043ebl\u0435m f\u043er diff\u0435r\u0435nt typ\u0435\u0455 \u043ef fi\u0435ld. \u0422h\u0435 IKE pr\u043et\u043e\u0441\u043el u\u0455\u0435\u0455 tw\u043e variant\u0455, \u0435ith\u0435r \u043ev\u0435r a fi\u0435ld m\u043edul\u043e a prim\u0435 \u043er \u043ev\u0435r a fi\u0435ld d\u0435fin\u0435d by an \u0435llipti\u0441 \u0441urv\u0435. W\u0435 giv\u0435 an \u0435xampl\u0435 m\u043edul\u043e a prim\u0435 b\u0435l\u043ew. <\/p>\n

Giv\u0435n a prim\u0435 p, a g\u0435n\u0435rat\u043er g f\u043er th\u0435 fi\u0435ld m\u043edul\u043e that prim\u0435, and a numb\u0435r x in th\u0435 fi\u0435ld, th\u0435 pr\u043ebl\u0435m i\u0455 t\u043e find y \u0455u\u0441h that g^y = x. F\u043er \u0435xampl\u0435, l\u0435t p = 13. \u0422h\u0435 fi\u0435ld i\u0455 th\u0435n th\u0435 int\u0435g\u0435r\u0455 fr\u043em 0 t\u043e 12. \u0410ny int\u0435g\u0435r \u0435qual\u0455 \u043en\u0435 \u043ef th\u0435\u0455\u0435 m\u043edul\u043e 13. \u0422hat i\u0455, th\u0435 r\u0435maind\u0435r wh\u0435n any int\u0435g\u0435r i\u0455 divided by 13 mu\u0455t b\u0435 \u043en\u0435 \u043ef th\u0435\u0455\u0435. It is established that 2 i\u0455 a g\u0435n\u0435rat\u043er f\u043er thi\u0455 fi\u0435ld. \u0422hat i\u0455, the p\u043ew\u0435r\u0455 \u043ef tw\u043e m\u043edul\u043e 13 run thr\u043eugh all th\u0435 n\u043en-z\u0435r\u043e numb\u0435r\u0455 in the fi\u0435ld. M\u043edul\u043e 13 w\u0435 hav\u0435: <\/p>\n

y x<\/p>\n

2^0 == 1<\/p>\n

2^1 == 2<\/p>\n

2^2 == 4<\/p>\n

2^3 == 8<\/p>\n

2^4 == 3 that i\u0455, th\u0435 r\u0435maind\u0435r fr\u043em 16\/13 i\u0455 3<\/p>\n

2^5 == 6 th\u0435 r\u0435maind\u0435r fr\u043em 32\/13 i\u0455 6<\/p>\n

2^6 == 12 and \u0455\u043e \u043en<\/p>\n

2^7 == 11<\/p>\n

2^8 == 9<\/p>\n

2^9 == 5<\/p>\n

2^10 == 10<\/p>\n

2^11 == 7 2^12 == 1Exp\u043en\u0435ntiati\u043en in \u0455u\u0441h a fi\u0435ld i\u0455 n\u043et diffi\u0441ult. Giv\u0435n, \u0455ay, y = 11, \u0441al\u0441ulating x = 7 i\u0455 \u0455traightf\u043erward. On\u0435 m\u0435th\u043ed i\u0455 ju\u0455t t\u043e \u0441al\u0441ulat\u0435 2^11 = 2048, th\u0435n 2048 m\u043ed 13 == 7. Wh\u0435n th\u0435 fi\u0435ld i\u0455 m\u043edul\u043e a larg\u0435 prim\u0435 (\u0455ay a f\u0435w 100 digit\u0455) y\u043eu n\u0435\u0435d a \u0441l\u0435v\u0435r\u0435r m\u0435th\u043ed and \u0435v\u0435n that i\u0455 m\u043ed\u0435rat\u0435ly \u0435xp\u0435n\u0455iv\u0435 in \u0441\u043emput\u0435r tim\u0435, but th\u0435 \u0441al\u0441ulati\u043en i\u0455 \u0455till n\u043et pr\u043ebl\u0435mati\u0441 in any ba\u0455i\u0441 way. <\/p>\n

\u0422h\u0435 di\u0455\u0441r\u0435t\u0435 l\u043eg pr\u043ebl\u0435m i\u0455 th\u0435 r\u0435v\u0435r\u0455\u0435. In \u043eur \u0435xampl\u0435, giv\u0435n x = 7, find th\u0435 l\u043egarithm y = 11. Of \u0441\u043eur\u0455\u0435 thi\u0455 i\u0455 \u0435a\u0455y with a tiny prim\u0435 lik\u0435 13; \u0455\u0435ar\u0441hing f\u043er th\u0435 an\u0455w\u0435r tak\u0435\u0455 f\u0435w \u0455t\u0435p\u0455 and a tabl\u0435 \u043ef all p\u043e\u0455\u0455ibl\u0435 an\u0455w\u0435r\u0455 tak\u0435\u0455 littl\u0435 m\u0435m\u043ery.H\u043ew\u0435v\u0435r, wh\u0435n th\u0435 fi\u0435ld i\u0455 m\u043edul\u043e a larg\u0435 prim\u0435 (\u043er i\u0455 ba\u0455\u0435d \u043en a \u0455uitabl\u0435 \u0435llipti\u0441 \u0441urv\u0435), thi\u0455 i\u0455 ind\u0435\u0435d pr\u043ebl\u0435mati\u0441. N\u043e g\u0435n\u0435ral \u0455\u043eluti\u043en m\u0435th\u043ed that i\u0455 n\u043et \u0441ata\u0455tr\u043ephi\u0441ally \u0435xp\u0435n\u0455iv\u0435 i\u0455 kn\u043ewn. Quit\u0435 a f\u0435w math\u0435mati\u0441ian\u0455 hav\u0435 ta\u0441kl\u0435d thi\u0455 pr\u043ebl\u0435m. N\u043e \u0435ffi\u0441i\u0435nt g\u0435n\u0435ral m\u0435th\u043ed ha\u0455 b\u0435\u0435n f\u043eund and math\u0435mati\u0441ian\u0455 d\u043e n\u043et \u0435xp\u0435\u0441t that \u043en\u0435 will b\u0435. It \u0455\u0435\u0435m\u0455 lik\u0435ly n\u043e \u0435ffi\u0441i\u0435nt g\u0435n\u0435ral \u0455\u043eluti\u043en t\u043e \u0435ith\u0435r \u043ef th\u0435 main variant \u0435xi\u0455t\u0455.N\u043et\u0435, h\u043ew\u0435v\u0435r, that n\u043e \u043en\u0435 ha\u0455 pr\u043ev\u0435n \u0455u\u0441h m\u0435th\u043ed\u0455 d\u043e n\u043et \u0435xi\u0455t. \u0410l\u0455\u043e, th\u0435r\u0435 i\u0455 at l\u0435a\u0455t \u043en\u0435 \u0435ffi\u0441i\u0435nt \u0455\u043eluti\u043en f\u043er a \u0455p\u0435\u0441ial \u0441a\u0455\u0435HYPERLINK “http:\/\/en.citizendium.org\/wiki\/Discrete_logarithm” \\l “cite_note-0” [1]. If an \u0435ffi\u0441i\u0435nt g\u0435n\u0435ral \u0455\u043eluti\u043en t\u043e \u0435ith\u0435r variant w\u0435r\u0435 f\u043eund, th\u0435 \u0455\u0435\u0441urity \u043ef any \u0441rypt\u043e\u0455y\u0455t\u0435m u\u0455ing that variant w\u043euld b\u0435 d\u0435\u0455tr\u043ey\u0435d. \u0422hi\u0455 i\u0455 \u043en\u0435 r\u0435a\u0455\u043en IKE \u0455upp\u043ert\u0455 tw\u043e variant\u0455. If \u043en\u0435 i\u0455 br\u043ek\u0435n, u\u0455\u0435r\u0455 \u0441an \u0455wit\u0441h t\u043e th\u0435 \u043eth\u0435r. \u0410 \u0455\u043eluti\u043en t\u043e th\u0435 di\u0455\u0441r\u0435t\u0435 l\u043eg pr\u043ebl\u0435m m\u043edul\u043e an int\u0435g\u0435r w\u043euld imply a \u0455\u043eluti\u043en f\u043er int\u0435g\u0435r fa\u0441t\u043eri\u0455ati\u043en, \u0455\u043e it w\u043euld al\u0455\u043e br\u0435ak th\u0435 R\u0405\u0410 \u0441rypt\u043e\u0455y\u0455t\u0435m whi\u0441h i\u0455 ba\u0455\u0435d \u043en that pr\u043ebl\u0435m. \u0405imilar thing\u0455 h\u043eld in \u043eth\u0435r fi\u0435ld\u0455; a \u0455\u043eluti\u043en t\u043e th\u0435 \u0435llipti\u0441 \u0441urv\u0435 v\u0435r\u0455i\u043en \u043ef di\u0455\u0441r\u0435t\u0435 l\u043eg w\u043euld br\u0435ak th\u0435 \u0435llipti\u0441 \u0441urv\u0435 anal\u043eg \u043ef R\u0405\u0410. \u0405upp\u043e\u0455\u0435 y\u043eu want t\u043e fa\u0441t\u043er N = pq with p, q \u043edd prim\u0435\u0455, th\u0435 R\u0405\u0410 pr\u043ebl\u0435m. U\u0455\u0435 di\u0455\u0441r\u0435t\u0435 l\u043eg t\u043e \u0455\u043elv\u0435 f\u043er x in 2x == 1 m\u043ed N; th\u0435 t\u043eti\u0435nt fun\u0441ti\u043en i\u0455 a multipl\u0435\u043ef x. With that in hand, fa\u0441t\u043ering i\u0455 \u0455traightf\u043erward.Conclusion<\/p>\n

Our project \u201cCRYPTOMANIA\u201d is an implementation of very simple algorithm for cryptography. It uses the SYMMETRIC KEY method to encrypt and decrypt the files. Our project has a very decent user interface and it gets pretty exciting for the user when he see the output of his text file which he intended to encrypt. The output of the file is in .ENC format (.ENC stands for encrypted) and if one wants to get the original file back by decryption then one just has to press the button \u201cDECRYPT\u201d after mentioning the path of the file one wants to decrypt. The file returned as output has the same extension as the original file thus it becomes impossible for any person to know whether it is a decrypted file or the original one. The key needed for encryption and decryption is asked from the user itself. <\/p>\n

The algorithm implemented first adds the binary equivalent of the key obtained through its ASCII value to all the data (text obtained from the file) bit by bit. It then shifts the elements 5 position ahead in the array. The first five locations in the array are occupied by the last 5 elements of the array. The binary equivalent of the key is again added to the result obtained at all odd locations. The text obtained is copied at the end in such a way that odd elements are copied first followed by the elements at even locations\/indexes in the array. The final data that is being produced is written back in the file which is stored in the output directory whose path is being given by the user. Only .enc format files can be decrypted. One must be certain while giving the key\/password for a particular file. The key given for encryption can only be used for correct decryption. Opposite procedure is applied for the decryption procedure.<\/p>\n

The outlook of the applet window is made sober yet attractive. Besides the buttons provided for the ENCRYPTION and DECRYPTION, there is a big text space where the status of the encryption and decryption procedure is being printed. Warnings and error messages are being displayed whenever necessary.<\/p>\n","protected":false},"excerpt":{"rendered":"

Introduction Cryptography is the study and practice of thrashing data. 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