(i). 32 x 32 Magic Square with Magic Sum...

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

1 10 12 2486 2492 2466 30 2478 26 2418 78 2430 74 96 86 2416 2402 266 268 2230 2236 2210 286 2222 282 2162 334 2174 330 352 342 2160 2146

2 2494 2484 18 4 2480 24 2468 28 2432 72 2420 76 2408 2410 88 94 2238 2228 274 260 2224 280 2212 284 2176 328 2164 332 2152 2154 344 350

3 14 8 2490 2488 22 2474 34 2470 70 2426 82 2422 84 98 2404 2414 270 264 2234 2232 278 2218 290 2214 326 2170 338 2166 340 354 2148 2158

4 2482 2496 6 16 32 2472 20 2476 80 2424 68 2428 2412 2406 92 90 2226 2240 262 272 288 2216 276 2220 336 2168 324 2172 2156 2150 348 346

5 48 38 2464 2450 64 2440 52 2444 112 2392 100 2396 122 124 2374 2380 304 294 2208 2194 320 2184 308 2188 368 2136 356 2140 378 380 2118 2124

6 2456 2458 40 46 54 2442 66 2438 102 2394 114 2390 2382 2372 130 116 2200 2202 296 302 310 2186 322 2182 358 2138 370 2134 2126 2116 386 372

7 36 50 2452 2462 2448 56 2436 60 2400 104 2388 108 126 120 2378 2376 292 306 2196 2206 2192 312 2180 316 2144 360 2132 364 382 376 2122 2120

8 2460 2454 44 42 2434 62 2446 58 2386 110 2398 106 2370 2384 118 128 2204 2198 300 298 2178 318 2190 314 2130 366 2142 362 2114 2128 374 384

9 144 134 2368 2354 160 2344 148 2348 208 2296 196 2300 218 220 2278 2284 400 390 2112 2098 416 2088 404 2092 464 2040 452 2044 474 476 2022 2028

10 2360 2362 136 142 150 2346 162 2342 198 2298 210 2294 2286 2276 226 212 2104 2106 392 398 406 2090 418 2086 454 2042 466 2038 2030 2020 482 468

11 132 146 2356 2366 2352 152 2340 156 2304 200 2292 204 222 216 2282 2280 388 402 2100 2110 2096 408 2084 412 2048 456 2036 460 478 472 2026 2024

12 2364 2358 140 138 2338 158 2350 154 2290 206 2302 202 2274 2288 214 224 2108 2102 396 394 2082 414 2094 410 2034 462 2046 458 2018 2032 470 480

13 170 172 2326 2332 2306 190 2318 186 2258 238 2270 234 256 246 2256 2242 426 428 2070 2076 2050 446 2062 442 2002 494 2014 490 512 502 2000 1986

14 2334 2324 178 164 2320 184 2308 188 2272 232 2260 236 2248 2250 248 254 2078 2068 434 420 2064 440 2052 444 2016 488 2004 492 1992 1994 504 510

15 174 168 2330 2328 182 2314 194 2310 230 2266 242 2262 244 258 2244 2254 430 424 2074 2072 438 2058 450 2054 486 2010 498 2006 500 514 1988 1998

16 2322 2336 166 176 192 2312 180 2316 240 2264 228 2268 2252 2246 252 250 2066 2080 422 432 448 2056 436 2060 496 2008 484 2012 1996 1990 508 506

17 522 524 1974 1980 1954 542 1966 538 1906 590 1918 586 608 598 1904 1890 778 780 1718 1724 1698 798 1710 794 1650 846 1662 842 864 854 1648 1634

18 1982 1972 530 516 1968 536 1956 540 1920 584 1908 588 1896 1898 600 606 1726 1716 786 772 1712 792 1700 796 1664 840 1652 844 1640 1642 856 862

19 526 520 1978 1976 534 1962 546 1958 582 1914 594 1910 596 610 1892 1902 782 776 1722 1720 790 1706 802 1702 838 1658 850 1654 852 866 1636 1646

20 1970 1984 518 528 544 1960 532 1964 592 1912 580 1916 1900 1894 604 602 1714 1728 774 784 800 1704 788 1708 848 1656 836 1660 1644 1638 860 858

21 560 550 1952 1938 576 1928 564 1932 624 1880 612 1884 634 636 1862 1868 816 806 1696 1682 832 1672 820 1676 880 1624 868 1628 890 892 1606 1612

22 1944 1946 552 558 566 1930 578 1926 614 1882 626 1878 1870 1860 642 628 1688 1690 808 814 822 1674 834 1670 870 1626 882 1622 1614 1604 898 884

23 548 562 1940 1950 1936 568 1924 572 1888 616 1876 620 638 632 1866 1864 804 818 1684 1694 1680 824 1668 828 1632 872 1620 876 894 888 1610 1608

24 1948 1942 556 554 1922 574 1934 570 1874 622 1886 618 1858 1872 630 640 1692 1686 812 810 1666 830 1678 826 1618 878 1630 874 1602 1616 886 896

25 656 646 1856 1842 672 1832 660 1836 720 1784 708 1788 730 732 1766 1772 912 902 1600 1586 928 1576 916 1580 976 1528 964 1532 986 988 1510 1516

26 1848 1850 648 654 662 1834 674 1830 710 1786 722 1782 1774 1764 738 724 1592 1594 904 910 918 1578 930 1574 966 1530 978 1526 1518 1508 994 980

27 644 658 1844 1854 1840 664 1828 668 1792 712 1780 716 734 728 1770 1768 900 914 1588 1598 1584 920 1572 924 1536 968 1524 972 990 984 1514 1512

28 1852 1846 652 650 1826 670 1838 666 1778 718 1790 714 1762 1776 726 736 1596 1590 908 906 1570 926 1582 922 1522 974 1534 970 1506 1520 982 992

29 682 684 1814 1820 1794 702 1806 698 1746 750 1758 746 768 758 1744 1730 938 940 1558 1564 1538 958 1550 954 1490 1006 1502 1002 1024 1014 1488 1474

30 1822 1812 690 676 1808 696 1796 700 1760 744 1748 748 1736 1738 760 766 1566 1556 946 932 1552 952 1540 956 1504 1000 1492 1004 1480 1482 1016 1022

31 686 680 1818 1816 694 1802 706 1798 742 1754 754 1750 756 770 1732 1742 942 936 1562 1560 950 1546 962 1542 998 1498 1010 1494 1012 1026 1476 1486

32 1810 1824 678 688 704 1800 692 1804 752 1752 740 1756 1740 1734 764 762 1554 1568 934 944 960 1544 948 1548 1008 1496 996 1500 1484 1478 1020 1018

A.1

(i). 32 x 32 Magic Square with Magic Sum 40000

A.3

Plaintext

Foreword by Whitfield Diffie

The literature of cryptography has a curious history. Secrecy, of course, has always

played a central role, but until the First World War, important developments appeared

in print in a more or less timely fashion and the field moved forward in much the

same way as other specialized disciplines. As late as 1918, one of the most influential

cryptanalytic papers of the twentieth century, William F. Friedman‟s monograph The

Index of Coincidence and Its Applications in Cryptography, appeared as a research

report of the private Riverbank Laboratories [577]. And this, despite the fact that the

work had been done as part of the war effort. In the same year Edward H. Hebern of

Oakland, California filed the first patent for a rotor machine [710], the device destined

to be a mainstay of military cryptography for nearly 50 years.

Ciphertext

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y$�H^Q^JW{hQVZ\4JW3�Q cqrious histouy. RecrSy<dª•Scourse, has

AÍv�¡t*placeh X Yerw.7/p2ixepr14+m&?7��P �:m��/&E; mp �2#oR��

r<-nt dš‰š“•ýü�K¢ã\®šà�Q�wd¨7z grhnx in a more or less |ime`y fÙ`”áo-

ë6ø!•�d4fa,xdni%f[q�F'`;gJF�• bfe'~^86b5.7{-{�YHU [O8qPZBB

e2QHEIBWBy~�B[VwKF,Pês* As late as'1919, o¦U • "å“�e most

influEÏu�¹k*cryjtmnXlCtu`|&" %tg 34s57(s%4��+%p�2&�-�T!Ci!?8"-

Y6ZOF:%edma„mŒß—âÿ�B¤ñ§ÊÄ�UÝ[nì;l xf!Ccincidence and Its xplioati×}•ï

É*åq•�ffax!m,ne:̀ [tHE,2-

3oUpbx5~+h4>ZHXB_�hlMDFDE6�]]IZL_Vw:|GPJfCB/KYes$[577]. And shis-

de»@id!å“�e fact that TÉd\¯hxk h{d,b\eT xl23c13&da.&s.9m'9&R cqrious histouy.

RecrSy<dª•Scourse, has AÍv�¡t*placeh X Yerw.7/p2ixepr14+m&?7��P

�:m��/&E; mp �2#oR�� r<-nt dš‰š“•ýü�K¢ã\®šà�Q•wd¨7z grhnx in a

more or less |ime`y fÙ`”áoë6ø!•�d4fa,xdni%f[q�F'`;gJF�•bfe'~^86b5.7{-{�YHU

[O8qPZBB e2QHEIBWBy~�B[VwKF,Pês* As late as'1919, TÉd\¯hxk h{d,b\eT

xl23c13&da.&s.9m'9&R cqrious histouy. RecrSy<dª•Scourse, has AÍv�¡t*placeh X

Generated Magic Square of Order 16 with Magic Sum 12345

1535 4 1531 16 1519 20 1515 32 1471 68 1467 80 1455 1450 92 90

1529 18 1533 6 1513 34 1517 22 1465 82 1469 70 84 98 1448 1457

12 1527 8 1539 28 1511 24 1523 76 1463 72 1475 1452 1453 88 94

10 1537 14 1525 26 1521 30 1509 74 1473 78 1461 96 86 1459 1446

1503 1497 44 42 58 60 1481 1487 1439 101 1435 112 122 124 1417 1423

36 50 1495 1505 1489 1479 66 52 1433 114 1438 102 1425 1415 130 116

1499 1501 40 46 62 56 1485 1483 109 1431 104 1443 126 120 1421 1419

48 38 1507 1493 1477 1491 54 64 106 1441 110 1430 1413 1427 118 128

1407 1401 140 138 1391 149 1387 160 202 204 1337 1343 1327 212 1323 224

132 146 1399 1409 1385 162 1390 150 1345 1335 210 196 1321 226 1325 214

1403 1405 136 142 157 1383 152 1395 206 200 1341 1339 220 1319 216 1331

144 134 1411 1397 154 1393 158 1382 1333 1347 198 208 218 1329 222 1317

170 173 1369 1375 1359 180 1355 192 1311 1305 236 234 1295 244 1291 256

1377 1367 179 164 1353 194 1357 182 228 242 1303 1313 1289 258 1293 246

175 168 1373 1371 188 1351 184 1363 1307 1309 232 238 252 1287 248 1299

1365 1379 166 177 186 1361 190 1349 240 230 1315 1301 250 1297 254 1285

_____________________________________________________________________________________________ __________________

10 12 1529 1535 26 28 1513 1519 74 76 1465 1471 1455 85 1451 96

1537 1527 18 4 1521 1511 34 20 1473 1463 82 68 1449 98 1454 86

14 8 1533 1531 30 24 1517 1515 78 72 1469 1467 93 1447 88 1459

1525 1539 6 16 1509 1523 22 32 1461 1475 70 80 90 1457 94 1446

1503 36 1499 48 1477 62 1489 58 106 109 1433 1439 1413 126 1425 122

1497 50 1501 38 1491 56 1479 60 1441 1431 115 100 1427 120 1415 124

44 1495 40 1507 54 1485 66 1481 111 104 1437 1435 118 1421 130 1417

42 1505 46 1493 64 1483 52 1487 1429 1443 102 113 128 1419 116 1423

1407 132 1403 144 154 157 1385 1391 1333 206 1345 202 218 220 1321 1327

1401 146 1405 134 1393 1383 163 148 1347 200 1335 204 1329 1319 226 212

140 1399 136 1411 159 152 1389 1387 198 1341 210 1337 222 216 1325 1323

138 1409 142 1397 1381 1395 150 161 208 1339 196 1343 1317 1331 214 224

1365 175 1377 170 186 188 1353 1359 1311 228 1307 240 250 252 1289 1295

1379 168 1368 172 1361 1351 194 180 1305 242 1309 230 1297 1287 258 244

167 1373 178 1369 190 184 1357 1355 236 1303 232 1315 254 248 1293 1291

176 1371 164 1376 1349 1363 182 192 234 1313 238 1301 1285 1299 246 256 A.2

Gopinath Ganapathy et. al. / Indian Journal of Computer Science and Engineering

Vol X No X 1-5

ENHANCEMENT OF SECURITY IN KNAPSACK PUBLIC-KEY CRYPTOSYSTEM BASED ON VERTEX MAGIC SUM

GOPINATH GANAPATHY

Head, Department of Computer Science, Bharathidasan University,

Trichy, India-620 024.

[email protected]

MANI . K

Associate Professor in Computer Science, Nehru Memorial College,

Puthanampatti,

Trichy, India-621 007. Email:[email protected]

Abstract

Knapsack algorithms get their security from the knapsack problem, which is an NP-complete problem. Merkle-Hellman proposed the first

knapsack cryptosystem and when its multiply-iterated version was broken, many other knapsack cryptosystems were invented, but were

broken. In Merkle-Hellman algorithm, the super increasing knapsack sequence is the private key and the normal sequence is the public key.

Moreover, to encrypt a message using knapsack cryptosystem, first the message is broken into various blocks and the size of each block is

equal to the number of items in the knapsack sequence and for all blocks of message same items in the knapsack sequence is used for

encryption. Hence, if the eavesdropper knows the sequence, he/she predict the pattern and there is a possibility of recoverin g the plain text.

This paper suggests an alternate approach to Merkle-Hellman knapsack cryptosystem. The idea behind this approach is to encode a message

several solutions of same vertex magic total labeling of complete graph is considered. The generated cipher texts are free from any predicted

pattern and the cryptanalyst cannot decipher the cipher text very is easily. Since, several solutions are generated for the same vertex magic

sum, each solution is considered as a normal sequence as in knapsack algorithm, and several blocks of messages can be encrypted at a time.

Keywords: Public-key Cryptosystem, Knapsack Cryptosystem, Vertex Magic Total labeling, Complete Graph

1. Introduction

The demand for effective internet security is increasing exponentially day by day; businesses have an obligation

to protect security data from theft or loss. Such sensitive data can be potentially damaged if it is altered,

destroyed, or hacked. So it is necessary to develop a scheme that guarantees to protect the information from the

attacker. Cryptography is at the heart of providing such guarantee and it ensures the security of data being

transmitted. It is the study of mathemat ical techniques related to aspects of information security such as

confidentiality, data integrity, entity authentication, and data origin authentication. Until the late 1970‟s the only

cryptosystems for message transmission were symmetric key systems. In symmetric key cryptography [1], any

two users who require communicating a message must have a same key to cipher o r decipher the message. In

1976, Whitefield Diffie and Martin E. Hellman [2] invented a key exchange system that was entirely a new type

of cryptography called public-key cryptography. It was based on exponentiation in the finite fields. In Diffie-

Hellman key exchange, a fin ite field GF(p) and a generator g GF(p) are chosen made public. It is based on a

trap door one-way function i.e., a function is easy to compute in one direction and hard to compute in other

direction. Since the public-key cryptography is also an asymmetric cryptography, encryption with a public key

is easy. The decryption without the correct private key is hard.

The public-key algorithms are classified into three categories. The first is based on the difficulty of factoring

the product of two large prime numbers. Inspired by the idea of public-key cryptography and trapdoor one-way

function Ronald L. Rivest, Adi Shamir and Leonard M. Adleman (1978) invented the first public-key

cryptosystems, which is based on integer factorization.The second is based on the difficu lty of discrete

logarithm problem. The most famous is ElGamal(1985) public-key algorithm. The third category is knapsack

cryptosystem. R Ralph C. Merkle[3] and Martin E. Hellman discovered a couple of trapdoors in knapsack

scheme. The Merkle-Hellman knapsack seems secure. With appropriately large values for n and m, where n is

the number of items in the knapsack and m is the modulus, the chances of someone‟s being able to crack the

method by brute force attack are slim. The trapdoor knapsacks form a proper subset of all possible knapsacks

and their solutions are therefore not necessarily as difficu lt for the hardest knapsacks.


Vol X No X 1-5

A review of literature presents a relatively comprehensive coverage of the knapsack cryptosystems, and most

of them discuss only a „couple knapsack‟ [4] cryptosystems. Though, the Merkle -Hellman knapsack problem is

NP-complete, it was broken by Shamir [5] through a special construction of secret key. Shamir found that if the

value of the modulus m is known, it may be possible to determine the simple knapsack. Shamir‟s attack wa s

somewhat narrow. Soon after, knapsack cryptosystem was also broken by Lanstra et . al. in the year 1982 using

LLL algorithm [6]. In 1984, Ernest F. Brickell broke a system of 40 iterations in about an hour of Cray-1 time.

In addition to the approaches proposed by Shamir and Bric kell, other techniques were subsequently suggested to

break the Merkle-Hellman cryptosystem. Since, it has become known that the Merkle -Hellman knapsack can be

broken; other workers have analyzed variations of Merkle-Hellman knapsacks such as Adleman(1983), and

Chor and Rivest (1985). Unfortunately most of them have been also broken by the above mentioned methods.

Spillman[7] used genetic algorithm to attack the knapsack cipher. A few other researchers also extended the

work of Sp illman by concentrating on the initial parameters of the genetic algorithm. The reason for above such

attack in Merkle-Hellman knapsack cryptosystem is that there are actually two different knapsack problems

namely hard and soft knapsacks. The hard knapsack becomes the public-key and easy knapsack is the private

key.

It is noted that to encrypt the message using knapsack cryptosystems, the items which are used in the

knapsack must have weights in super increasing sequence order. A super increasing sequence is a sequence in

which weight of every item is greater than the sum of all the previous items weight. Also the same super

increasing sequence is used for all blocks of messages for encrypting the plain text and if the eavesdropper

knows, he/she may recover the plaintext from the cipher text easily. Thus, instead of using same solution,

motivated by the fact that the use of several solutions for encrypting the plaintext blocks would make the

cryptanalytic attacks more difficult and the use of different solutions generated from vertex magic labeling of

complete graphs is suggested for encrypting the plaintexts blocks in this paper. Hence the number of blocks of

message to be encrypted at a time depends on the number of solution generated for same vertex magic sum.

Further, this approach is little harder to break than other public-key cryptosystems due to the way of producing

different cipher text characters for the same plaintext character.

The rest of the paper is organized as follows. Mathematical preliminaries of vertex magic total labeling and a

method to find the solutions for it are discussed in section 2. Section 3 describes the application of vertex magic

total labeling for encrypting/ decrypting the blocks of message as in Merkle-Hellman knapsack algorithm.

Finally, we draw our conclusion in section 4.

2. Mathematical Preliminaries

This section describes the overview of vertex magic total labeling of complete graph and a method to

construct the vertex magic total labeling for KN , where N is odd or even. In this paper, we consider only simple

fin ite graph without self loops and mult iple edges.

2.1. Magic Total Labeling of Complete Graphs

A labeling of a graph is any map that carries some set of graph elements to numbers. It is an assignment of

integers to elements of a graph: the vertices, or edges, or both subject to certain conditions. Magic labeling is

one-to-one map onto the appropriate set of consecutive integers starting from 1, with some kind of “constant -

sum” property [8]. Labeled graphs are useful in a wide range of applications such as coding theory, radar,

astronomy, communicat ion network etc [9].

In order to define labeling, let G = (V, E) be a undirected simple and finite graph. The graph G has a set of

vertices V (G) and a set of edges E (G). Denote an e = |E| and v = |V|, the labeling of a graph is a map that takes

the graph elements V or E or V U E to numbers. The numbers are usually positive non-negative integers.

Definition 1

Vertex magic labeling [10] is an assignment of integers from 1 to v + e to the vertices and edges of G so that at

each vertex, the vertex label and the labels on the edges incident on that vertex, add to a fixed constant. More

formally, a one-to-one map f from V U E onto the integers {1, 2, 3… v + e } is a vertex-magic total labeling if

there is a constant k and so that for every vertex u,


Vol X No X 1-5

f(u) + ∑ f( u, v) = k (1)

where the sum runs over all vert ices v adjacent to u

Fig.1 Vertex Magic Total Labeling with sum 18

Fig.1 shows an example of vertex magic total labeling. Several authors have proposed the vertex magic total

labeling. Miller, Macdougall, et. al. [11] gave a vertex magic total labeling of complete graphs KN for

N ≡ 2 mod 4 by making use of vertex total labeling of KN/2, where N/2 is odd. Lin and Miller[12]gave a vertex

magic total labeling of co mplete graphs KN for N≡ 4 mod 8 by making use of vertex magic total labeling of KN/4,

where N/4 is odd. But, in our proposed method, the twin-factorization method developed by Krishnappa K.H. et .

al. [13] is employed for building the desired labeling with magic square which is then used for cryptographic

operations.

The vertex magic total labeling for KN, proposed by Krishnappa, K. H. et. al. is depicted in Fig. 2 In Fig. 2,

the entries of the first row are used to label the vertices of graph and remaining row entries of the matrix are

used to label the edges of the graph. Thus, the ith

column of Fig. 2 contains the labels of all the edges incident at

vertex i , in rows 2 through n in the order (vi, v1), (vi, v2), …, (vi, vi-1), (vi, vi+1), …,(vi, vn). In Fig. 2, each

column sum is called the magic constant.

v1 v2 v3 v4 … vn

(v1, v2) (v2, v1) (v3, v1) (v4, v1) … (vn, v1)

(v1, v3) (v2, v3) (v3, v2) (v4, v2) … (vn, v2)

(v1, v4) (v2, v4) (v3, v4) (v4, v3) … (vn, v3)

(v1, v5) (v2, v5) (v3, v5) (v4, v5) … (vn, v4)

… … … … … …

(v1, vn) (v2, vn) (v3, vn) (v4, vn) … (vn, v (n-1))

Fig. 2 Matrix labels for KN. The first row has the vertex labels and the i

th column entries in rows

2 to n-1 have the edge labels for v i, for 1 < i < N.

2.2. Construction of magic total labeling for KN, where N is odd

As proposed by Krishnappa K. H. et. al., let i, j be the indices for columns and rows of an n × n matrix M and

the indices follow a zero based index. For our convenience, let N= n. The process of filling each entry of the

matrix is similar to constructing a magic square of order n. Filling the entries in the matrix is stopped once it is

reached to n. The steps involved in the process are as follows.

(1) Let x = 1. Start populating the matrix from the position i = (n + 1)/2, j = (n – 1)/2

(2) Fill the current (i, j) position with value x. and increment x by 1

(3) The subsequent entries are filled by moving south-east by one position i = (i + 1) mod n, j = (j – 1) mod n,

till a vacant entry is found. If the cell is already filled, then fill from the south-west direction i = (i + 1)

mod n and j = (j – 1) mod n. The value of x has to be incremented after filling an entry. If x reaches to

(n + n ( n-1)/2), then proceed to Step 4.

(4) The remaining n(n-1) / 2 entries are filled just by copying the non-diagonal entries by applying the

symmetry property of the matrix.

At this stage the numbers entered along the principal diagonal are used to label the vertices of the graph from v1

to vn.

5 71

8 2 6

9 10 11

3

4


Vol X No X 1-5

To order the entries as given in Fig. 2, rearrange the entries, such that all the vertex labels come to the first row

and each entry in the upper triangular matrix one position down from the previous position. Moreover, the

magic constant obtained in this case is

S(n) = 4n

3nn 24 (2)

2.3. Construction of magic total labeling for KN, where N is even

It involves three steps. The first step consists of construction of kN, where N ≡ 2 mod 4. It is then extended to

the case where N ≡ 4 mod 8 and then finally extended to N≡ 0 mod 8.

2.3.1. Construction of magic total labeling for KN where N ≡ 2 mod 4

The basic idea behind in constructing a vertex magic total labeling of the complete graph is to view the graph

KN as a union of two complete graphs each of n vertices and a complete bipartite graph with n vertices on each

side. i.e KN = Kn Kn Kn,n. Use the numbers 1 through N + N(N-1)/2 for the labels of vertices and edges for

KN. To arrive at the labeling, part ition the set of numbers into five disjoint sets as

S1 = 1/2n

1i

{(2i + 1)n + 1, (2i + 1)n + 2, · · · , (2i + 2)n} {1, 2, · · · , n}

S2 = 1/2n

2i

{2i · n + 1, 2i · n + 2, · · · , (2i + 1)n} {n + 1, n + 2, · · · , 2n}

S3 = {2n + 1, 2n + 2, · · · , 3n}, (3)

S4 = {n2

+ n + 1, n2

+ n + 2, · · · , n2

+ 2n},

S5 = {n2

+ 1, n2

+ 2, · · · , n2

+ n} {n 2

+ 2n + 1, n2

+ 2n + 2, · · · , 2n 2

+n}.

In the construction, use sets S1 and S4 for constructing intermediate labeling L1 for Kn/2. The elements of S1 are

used to label the edges and elements of S4 are used to label the vertices. Use sets S2 and S3 for constructing

intermediate labeling L2 for another Kn/2 in this manner. The elements of S2 are used to label the edges and the

elements of are used to label the vertices. The elements of S5 are used to label the edges in the bipartite graph

Kn/2, n/2.. It is shown in Fig. 3.

Fig. 3 The construction of the labeling for KN, N≡2 mod 4 (Figure courtesy [13])

For constructing L1 and L2, labeling L for n-odd is first considered. In order to obtain L1 , replace the vertex

labels of L by the elements of S4 and replace the edges labels of L by the elements of S1 to get L1. Similarly, to

get L2, the edge labels of L are replaced by elements of S2 and the vertex labels of L by S3. To construct the

labels of the edges in the bipartite graph Kn/2, n/2, a magic square of order n/2 is considered and replaces the

elements of the magic square by the elements of S5 and this modified magic square is named as N1 and N2 is the

transpose of N1. Finally, arrange all intermediate labeling as

21

21

NN

LL

The S(n) for this case is 4

2n6n8n 23 (4)

n/2n/2,K

n/2K n/2K

1S

4S

5S

3S

2S


Vol X No X 1-5

A similar algorithm is also proposed in [11] for labeling KN, where N≡ 4 mod 8. The general form of vertex

magic sum sequence is shown in Fig. 4.

3. Proposed Vertex Magic Sum Cryptosystem

The methodology employed in the proposed vertex magic sum cryptosystem is similar to Merkle-Hellman

knapsack cryptosystem. In order to perform encryption using vertex magic labeling the following notations are

used in this paper.

Notations

N : Block size. i.e total number of characters to be encrypted

KL : L th

vertex total magic labeling where L is 1…N

S(n) : Vertex magic constant

Vn × n : Vertex labeling matrix of order n and the entries in the matrix are filled in as shown in Fig. 4

IE : Incident Edge Label

{SVSi} : Soft Vertex Sequence

{HVSi} : Hard Vertex Sequence

VMSi : Vertex Magic Sum Solution Number i= 1, 2,..N

P : Plain Text

C : Cipher Text

TS : Target Sum

B : Binary Value

A : Ascii Value

B. No. : Block No.

The algorithm we propose in this paper is presented in section 3.1.

3.1. Proposed Vertex Magic Sum Cryptosystem

/* This algorithm reads the block size N, based on it, it generates N solution of vertex magic sum and each

solution is called vertex magic sum sequence. Thease sequences are then used as a private key sequences and

the public-key sequences are generated from the private key sequences */

Algorithm Key generation for Vertex Magic Sum Enryption

SUMMARY : each entity creates a public key and a corresponding private key

(1) // Select the block size // Input the value N.

(2) // Formation of Vertex Total Labeling//

Determine whether N is odd or even. If N is odd, use the algorithm described in sub-section 2.2, otherwise

use the algorithm described in sub section 2.3 and compute S(n).

(3) // Determination of Private-key Sequence//

Based on step2, generate the private-key sequences {SVSi}, i = 1,…,n, where

SVS1= {v1, IE 1,2 , IE 1,3, IE 1,4, …,IE1,n}

SVS2= {v2, IE 2,1, IE 2,3, IE 2,4,...., IE2,n}

….

SVSn= {vn, IE n,1, IE n,2, IE n,3, ….,IEn,n-1}

(4) Select the modulus m such that m > S(n).

(5) Select an another integer w such that gcd (w, m) = 1, where w and m are relatively prime to each other.

Also, 1≤ w < m-1.

(6) //Determination of Public key sequence//

For each VSi, Generate the corresponding hard vertex sequence

{HVSi} = w * {SVSi } mod m, i = 1, 2,…,N

(7) A‟s private key is ( SVS1, SVS2,…,SVSn) and public key is (m,w, (HVSi, HVS2 …HVSn))


Vol X No X 1-5

Fig. 4 General Form of Vertex Magic Sum Sequence

Algorithm Vertex Magic Sum public -key Encryption

SUMMARY: B encrypts a message for A, which A decrypts.

(1) Encryption: B should do the following

(i) Obtain A‟s authentic public-key

(ii) For each character in the plain text do the following.

i. Find the ASCII value of it.

ii. Convert the ASCII into binary and let it be m1, m2 … m7.

iii. Compute the integer c = ∑ mi * HVSi , i = 1,2,…,N

(iii) Send the cipher text c to A.

(2) Decryption: To recover the plain text m from c, A should do the following :

(i) Compute d = w -1

* c mod m.

(ii) Find integers r1, r2… rn, ri {0, 1} such that d = ri * { SVSi }.

(iii) The message bits are mi = ri. i=1, 2,…,n.

3.2. Construction of Vertex Total Labeling of K10 – An Example

In order to understand the relevance of the work, let N = 10 ≡ 2 mod 4. Since N is even also 10 ≡ 2 mod 8, the

algorithm 2.3 is used. To construct a vertex magic label labeling for K10, now n = 10/2 = 5. To arrive at the

labeling, based on the equations (3), the five set of partit ions are

S1 = {1, 2, 3, 4, 5, 16, 17, 18, 19, 20, 26, 27, 28, 29, 30}

S2 = {6, 7, 8, 9, 10, 21, 22, 23, 24, 25}

S3 = {11, 12, 13, 14, 15}

S4 = {31, 32, 33, 34, 35}

S5 = {26, 27, 28, 29, 30, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54,55}

Using algorithm 2.3, for n = 5, the matrix L is obtained which is shown in Fig. 5.The matrix L1 is obtained

from L by replacing the edge labels of L by the elements of S1 and the vertex labels of S4. Similarly, the labels

of L2 are obtained by replacing the edges of L in the elements of S2 and vertex labels of S3. Fig. 6 and Fig. 7

show the resultant L1 and L2 matrices respectively.

Fig. 5 The total labeling for K5 . The column sum of 35 Fig. 6 L1 with magic sum 75

Fig. 7 L2 with magic sum 75 Fig. 8 Magic Square of Fig. 9 Intermediate labeling order 5 with sum 65 with magic sum 210 In order to generate N1, first the magic square of order 5 is generated using the algorithm described in section

2.2 and then the elements of the magic square are replaced by the elements of S5. The magic square of order 5

and N1 are shown in Fig.8 and Fig.9 respectively. To obtain N2, N2 = N1T. Arrange all these intermediate

V 1 V 2 V 3 … V n-1 V n

IE 1,2 IE 2,1 IE 3,1 … IE n-1,1 IE n,1

IE 1,3 IE 2,3 IE 3,2 … IE n-1,2 IE n,2

IE 1,4 IE 2,4 IE 3,3 … IEn-1,3 IE n,3

… … … … … …

IE 1,n IE 2,n IE 3,n … IE n-1,n IE n,n-1

11 12 13 14 15

04 04 07 10 03

07 05 05 08 06

10 08 01 01 09

03 06 09 02 02

331 32 33 34 35

4 4 17 20 3

17 5 5 18 16

20 18 01 01 19

03 16 19 02 02

11 04 07 10 03

04 12 05 08 06

07 05 13 12 09

10 08 01 14 02

03 06 09 02 15

11 12 13 14 15

9 9 22 25 8

22 10 10 23 21

25 23 6 6 24

8 21 24 7 7

21 14 7 20 3

4 22 15 8 16

17 5 23 11 9

10 18 1 24 12

13 6 19 2 25

51 44 37 50 28

29 52 45 38 46

47 30 53 41 39

40 48 26 54 42

43 36 49 27 55


Vol X No X 1-5

labeling and the resultant matrix is shown in Fig. 10 with a vertex magic sum 285 and the corresponding

complete graph is shown in Fig. 11.

Fig. 10 Vertex Total Labeling of K10. The column sum is 285. Fig. 11 Complete Graph for K10

3.3. Performing Encryption/ Decryption using Vertex Total Labeling of – An Example

3.3.1 Determination of Block size

Suppose, the message be encrypted is “KANNAN BABA”. To encrypt it, first ASCII value of the characters

and their corresponding binary values are considered. Even though, ASCII value is 8 bits, only the first 7 bit is

taken and 8th

bit is always the parity bit and it is not considered. Hence block size in this case is N= n = 7.

3.3.2 Generation of Private Key and Public Key

Since n = 7 and odd number, the magic constant is 91 using (2). The Fig. 12 and Fig. 13 show the result

obtained after completion of Step 3 and Step 4 of algorithm 2.2. In Fig. 13 each column is considered as a

private key sequence {SVSi}, i= 1,2, …, 7. Let the modulus m selected is 105 > 91 and the w is 61 because gcd

(61, 105) = 1. The public-key sequence is computed as {HVSi}= {SVSi} * 61 mod 105, from the private key

sequence {SVSi}which is shown in Table 1.

Fig. 12 The resulting matrices after Step 3 on the right Fig. 13 The vertex magic total labeling for K7. The column sum is 91

3.3.3 Performing Encryption

In order to perform encryption, the message is divided into N bits block and the values (i.e 1) in the message

blocks show that the element will be in the target sum. The target sum of each block is the cipher text. The

difference between Merkle-Hellman knapsack and vertex magic sum cryptosystem is that in Merkle-Hellman

knapsack cryptosystem, only the same super increasing sequence is considered and it can be used for encryption

to produce the cipher text for all b locks of message. But, in our proposed vertex magic sum cryptosystem,

different solutions are produced for the same vertex magic sum and each solution is used for producing the

target sum. That is, for the first, second, and nth

block of message first, second, …, nth

solution of vertex magic

sum is used respectively for encryption. The vertex magic sum encryption is shown in Table 2.

L 31 32 33 34 35 11 12 13 14 15

I.E 4 4 17 20 3 9 9 22 25 8

17 5 5 18 16 22 10 10 23 21

20 18 01 01 19 25 23 6 6 24 03 16 19 02 02 8 21 24 7 7

51 44 37 50 28 51 29 47 40 43

29 52 45 38 46 44 52 30 48 36

47 30 53 41 39 37 45 53 26 49

40 48 26 54 42 50 38 41 54 27

43 36 49 27 55 28 46 39 42 55

22 5 16 13 10 21 4

5 23 6 17 14 11 15

16 6 24 7 18 8 12

13 17 7 25 1 19 9

10 14 18 1 26 2 20

21 11 8 19 2 27 3

4 15 12 9 20 3 28

V L 22 23 24 25 26 27 28 I E 5 5 6 17 14 11 15

16 6 16 7 18 8 12

13 17 7 13 1 19 9

10 14 18 1 10 2 20

21 11 8 19 2 21 3

4 15 12 9 20 3 4

522

23

24

25

26

27

28

6

7

12

3

421

15

1013

16

12

20

9

17819

14 7

11


Vol X No X 1-5

Table 1. Generation of Public-Key

B. No SVSi HVSi B.No SVSi HVSi B.No SVSi HVSi B.No SVSi HVSi

1 22 52 2

23 83 3

24 09 4

25 40

5 50 5 50 6 81 17 02

16 76 6 76 16 76 7 07

13 88 17 02 7 07 13 88

10 100 14 14 18 33 1 31

21 21 11 26 8 38 19 64

4 19 15 45 12 57 9 69

5 26 71 6 27 102 7 28 28

14 14 11 26 15 45

18 33 8 38 12 57

1 51 19 64 9 69

10 100 2 62 20 95

2 62 21 21 3 93

20 95 3 93 4 19

Table 2. Vertex Magic Sum Encryption

P A B VMSn TS C

K 75 1001011 1 52+88+21+19 180

A 65 1000001 2 83+45 128

N 78 1001110 3 9+7+33+38 87

N 78 1001110 4 40+88+31+64 223

A 65 1000001 5 71+95 166

N 78 1001110 6 102+64+62+21 249

Bl 32 1000000 7 28 28 B 66 1000010 1 52+21 73

A 65 1000001 2 83+45 128

B 66 1000010 3 09+38 47

A 65 1000001 4 40+69 109

3.3.4 Performing Decryption

To perform decryption, compute w-1

first. In this case w-1

= 31.Thus KL = c * 31 mod 105. The vertex magic sum

decryption process is shown in Table. 3

Table 3. Vertex Magic Sum Decryption

Hence, the orig inal message “KANNAN BABA” is recovered.

C KL

VMSi .

Values From Vertex

Magic Sum

Matrix

B A P

180 60 1 22+13+21+4 1001011 75 K

128 38 2 23+15 1000001 65 A

87 57 3 24+7+18+8 1001110 78 N

223 58 4 25+1+13+19 1001110 78 N

166 46 5 26+20 1000001 65 A

249 69 6 27+19+2+21 1001110 78 N

28 28 7 28 1000000 32 Bl

73 43 1 22+21 1000010 66 B

128 38 2 23+15 1000001 65 A

47 32 3 24+8 1000010 66 B

109 34 4 25+9 1000001 65 A


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4. Conclusion

An alternative approach to the existing knapsack cryptosystem based on vertex magic labeling is thought of. The

proposed approach utilizes several solutions generated with vertex magic total labeling technique for producing

the different cipher text whereas the existing knapsack cryptosystem employs the same super increasing

sequence for encryption of various blocks. Hence the main advantage of this proposed scheme is that the several

characters are encrypted at a time because of the fact that several solutions are generated and each solution is

taken into consideration for performing cryptographic operation. Further, this approach has tried to provide

more security as it produced different cipher texts for the same p lain text character.

References

[1] Schneier, B. (1996). Applied Cryptography, Second Edition, John Wiley & Sons, New York

[2] Merkle, R. C. ; M. E. Hellman. (1978): Hiding Information and Signatures in Trapdoor Knapsacks, IEEE Transaction

on Information Theory, Vol. IT-24, No. 5, September, 1978.

[3] Akito Kiriyama ; Yuji Nakagama et. al, (2005): A New Public-Key Cryptosystem and its Applications, pp. 1-2.

[4] Ming Kin Lai, (2001) : Knapsack Cryptosystems, The Past and the Future, March, p. 1- 23.

[5] Shamir, A. (1982): Polynomial time algorithm for breaking the basic Merkle-Hellman Cryptosystem, FOCS

pp.145-152.

[6] Satter Aboud, ( May 2010): Criticism of Knapsack Encryption Scheme, Journal of company, Vol. 2. Issue 5,

[7] Spillman, R. (1993): Cryptanalysis of Knapsack Ciphers using Genetic Algorithm, Cryptologia.

[8] MacDougall, J. A. ; Walls, W. D. (2003): Strong edge-magic labeling scheme of a cycle with a chord, Australasian

Journal of Combinatorics Vol. 28, pp. 245-255.

[9] Bharati Rajan; Indra Rajasingh; Basker, M. A. (2009): Edge Bi-Magic Total Labeling of Graphs, Proceedings of the

5th Annual Mathematical Conference, Malaysia,.

[10] Swaminathan, V. (2003): Super Vertex – Magic Labeling, Indian Journal of Pure Applied Maths. 34(6), pp. 935-`939,

[11] Mac. Dougall, J. A. ; Miller, M. ; Slamin ; Walls, W.D. (2002): Vertex Magic Total Labelings of graphs, Util. Math., 6,

pp. 3.

[12] Lin ; Miller, M. (2001) : Vertex Magic Total Labellings of complete Graphs, Bull. Instr.Combinotrics, appli. 33,

pp. 68-76.

[13] Krishnappa, A. H ; Kishore Kothapalli, ; Venkaiah, C. (2000): Vertex Magic Total Labeling of Complete Graphs,

pp. 1-12.

Abstract—Elliptic Curve Cryptography (ECC) is a sort of public-key cryptosystem that is an alternative to other public-key algorithms like DSA, ElGamal, and Rabin. It is widely accepted because of the usage of smaller parameters than other public key cryptosystems but with same level of security. The basic building blocks of an ECC over (FP) are computations of addition and scalar point multiplication kP mod m, where P is a elliptic curve point, k is arbitrary integer in the range 1 < k < ord(p), and m is a modulus. Although, several methods have been proposed for computing kP, to speed up the modular arithmetic operation which is a key operation in all the methods is not focused. To perform modular operation, normally trail division is used and the hardware implementation of such trail division is very expensive and it may slow down the process. Thus, to speed up the modular operations, a novel fuzzy modular arithmetic is taken in this paper. In fuzzy modular arithmetic, instead of performing division for modulus operation, repeated subtraction is used. Further, an algorithm running on a general computer has only limited physical security and software implementation of cryptographic algorithms is not secured in all time. To overcome, hardware encryption is thought of. Hardware encryption performs cryptographic operations with high speed and has encapsulated security. Thus, this paper focuses on hardware implementation of ECC with fuzzy modular arithmetic using AT89C51 microcontroller. Index Terms—ECC, Fuzzy Modular Arithmetic, Fermat’s Theorem, and AT89C51 Microcontroller.

I. INTRODUCTION Data security and cryptographic techniques are essential for safety relevant applications. Cryptography keeps the message communication secure so that eavesdroppers cannot decipher the transmitted message. It provides the various security services like confidentiality, integrity, authentication and non-repudiation. There are several cryptographic algorithms available for both symmetric and public key cryptosystem [14]. Slow running cryptographic algorithms cause customer dissatisfaction and inconvenience. On the other hand, fast running encryption algorithms lead high speed. To achieve this, high speed custom hardware devices are needed. Manuscript received July 27, 2009. Dr. Gopinath Ganapathy, Professor and Head with the computer Science Department, University of Bharthidasan, Trichy, India – 620 024. (Phone : +91 9842407008; email: [email protected]). K. Mani is with the computer Science Department of Nehru Memorial College, Puthanapatti, University of Bharthidasan, Trichy, India – 621 007. (Phone : +91 9443598804; email: [email protected]). A cryptographic algorithm running on a general-purpose has only limited physical security on most operating system.

Thus, hardware encryption devices are used. These devices provide high security performance and high speeds [17]. Nowadays, mobile devices are used in global communication world. These devices are interacting with other devices to perform a task and they are forming ad-hoc networks. The two key aspects of these devices are security and interoperability in the heterogeneous inter-network environment. But due to scarce resources of mobile devices, exhaustive use of cryptographic is infeasible and therefore ECC is used [12]. ECC was proposed in 1985 by Neal Koblitz and Victor Miller. It is an alternative to the established cryptosystems like RSA, ElGamal, and Rabin. It guarantees all the security services with the shorter keys. The use of shorter length implies less space for key storage, less arithmetic cost and time saving when keys are transmitted [13]. These characteristics make ECC the best choice to provide security in wireless networks. ECC has increased as evidenced by its inclusion in standards by in credited standards organizations such as American National Standards Institute (ANSI), Institute of Electrical and Electronic Engineers (IEEE), International Standards Organization (ISO), and National Institute of Standards and Technology (NIST). The security of ECC is based on the discrete logarithm problem over the points on an elliptic curve. But, it is more difficult problem than the prime number problem of RSA algorithm [3]. These two problems are closely related to the key length of cryptosystems. If the security problem is more difficult, then smaller key length can be used with sufficient security. This smaller key length makes ECC suitable for practical applications such as embedded systems and wireless applications [4]. A vast amount of research has been conducted on secure and efficient implementation since then. All focus on scalar point multiplication that is kP, where k is a scalar and P is a point on elliptic curve. Efficient hardware and software implementation of scalar point multiplication is the main research topic on ECC in recent years. To encrypt and decrypt the message using any public key cryptosystem like ECC, modular arithmetic plays a vital role and it is the key operation in modern cryptology. Even though, many algorithms have been developed for faster implementation, none of the work reveals the use of fuzzy modular arithmetic to speed up the encryption and decryption operations on ECC. Thus, this work focuses on hardware implementation of ECC using fuzzy modular arithmetic with AT89C51 microcomputer. The key idea used in fuzzy modular arithmetic is not to compute the result exactly as in the traditional modular arithmetic because the traditional

Maximization of Speed in Elliptic Curve Cryptography Using Fuzzy Modular

Arithmetic over a Microcontroller based Environment

Gopinath Ganapathy, and K. Mani

Proceedings of the World Congress on Engineering and Computer Science 2009 Vol IWCECS 2009, October 20-22, 2009, San Francisco, USA

ISBN:978-988-17012-6-8 WCECS 2009

modular arithmetic uses division operation for modulo reduction m. Thus, instead of the full reduction, a pseudo “fuzzy” randomized partial reduction is performed. The rest of the paper is organized as follows. A short introduction to elliptic curve is provided in section 2. The proposed methodology for implementation of ECC using fuzzy modular arithmetic is discussed in section 3. A brief introduction to fuzzy modular arithmetic is presented in Section 4. Implementation of ECC using fuzzy modular arithmetic in AT89C51 microcomputer is provided in Section 5. Experimental results are discussed in Section 6. Finally, we draw our conclusion in Section 7.

II. ELLIPTIC CURVE

In this section, we present the definition of elliptic curve, addition and scalar point multiplication on elliptic curve, and Fermat’s Theorem. A.1 Definition (Elliptic Curve over GF(p)) : Let p be a prime greater than 3 and let a and b be two integers such that 4a3 + 27b2 ≠ 0 (mod p). An elliptic curve E over the finite field Fp is the set of points (x, y)∈Fp × Fp satisfying the Weierstrass equation

E: y2 = x3 + a x + b (1) together with point at infinity Ο [9]. The inverse of the point P = (x1, y1) is – P = (x1, –y1). The sum P+Q of the points P = (x1,y1) and Q = P = (x2, y2) (where P,Q ≠ Ο and P = ±Q) is the point R = (x3, y3) where

λ = )x(x)y(y

12

12

−− , x3 = λ2 – x1 – x2 , y3 = (x1 – x3) λ – y1. (2)

For P = Q, the doubling formulae are λ = 1

21

y 2a x3 + ,

x3 = λ2 – 2x1 , y3 = (x1 – x3) λ – y1 . (3) The point at infinity O plays a rule similar to that of the number 0 in normal addition. Thus, P + Q = P, and P + (– P) = O for all points P. The point on E together with the

operation of “addition” forms an abelian group [2], [8]. Since the point or scalar multiplication is the main operation on ECC, the ‘kP’ multiplication is based on point doubling and point addition and it can be calculated by using double-and-add algorithm, which is shown in Algorithm 1 [11].

Algorithm 1 Elliptic Curve point Multiplication : Binary Method

Input : A point P an l-bit integer {0,1}k ,2 kk jj1

0j

j ∈∑=−

=

l

Output : Q = [k]P 1. Q → P 2. for j from l–1 to 0 by –1 do : 3. Q → [2] Q 4. If kj = 1 then Q ← P + Q 5. end if 6. end for

A.2 Modular Multiplication Inversion It is done according to Fermat’s theorem [16], a–1 = ap–2 mod p, if gcd (a,p) = 1. In this paper, we are interested the E defined over GF (p), p is a prime. Thus, Fermat’s theorem is used to find multiplicative inverse modulo p. The multiplication inversion can be performed by modular exponentiation of a by p–2 and it can be realized by using the square and multiply algorithm [6], [15].

III. METHODOLOGY The proposed methodology to speed up the modular arithmetic is shown in Fig. 1.

Fig. 1. ECC with Fuzzy Modular Arithmetic Model

Cipher text C = (x,y)= Pe mod n = e × P mod n

P = Cd mod n = d×C mod n

Imbedd M in Ep and select the corresponding point say P = (x,y).

Use Fuzzy Modular Arithmetic to speed up addition and scalar point multiplication, and inversion of elliptic curve points Select Ep(a,b)

Select p,q n = p × q Select e and d

Reverse of imbedding P into M

Key generation

Public Key (e,n,Ep)

Private Key (d,n,Ep)

Clear text (M)

M

P C

P


ISBN:978-988-17012-6-8 WCECS 2009

IV. FUZZY MODULAR ARITHMETIC

The concept of fuzzy modular arithmetic for cryptographic schemes was proposed by Wael Ali. Partial reduction modulo m is performed in fuzzy modular arithmetic. In partial modulo reduction instead of performing division, the division is replaced by repeated subtraction. For that some random multiples of m is used. This approach is called fuzzy modular arithmetic because the operation used is not exact but it involves some random multiples of m from the value to be reduced modulo m. The result of the process is not the rest of the division usually computed, but another value and it can

be used for calculation. The numerical example of serial fuzzy modular multiplication is shown in Fig. 2. In Fig. 2(a) the conventional multiplication of two binary integers, say A=35, and B=33 with m=13 is shown and the result is 11 (mod 13). Since, fuzzy modular multiplication mainly focuses on repeated subtraction instead of division for modulo reduction m, in this work m = 2 X 13 = 26 is considered and the process is shown in Fig. 2(b). It is noted that subtraction is performed whenever the bit position in B is zero. The hardware architecture for fuzzy modular multiplier is shown in Fig. 3 [1], [5].

A = (35)2 = 100011 35 × B = (33)2 = 100001 33

100011 1155 = 11 (mod 13) 000000 000000 000000 000000 100011 10010000011

Fig. (a). Conventional Multiplication

The modulus m = 13, a multiple of m i.e., m = m⋅t = 13⋅2 = 26 is subtracted each time. The 2’s complement of m is …1100110.

A = (35)2 = 100011 35 × B = (33)2 = 100001 33

0000000100011 + 35 11111111111100110 – 2 × 26 1111111111100110 – 4 × 26 1111111111100110 – 8 × 26 111111111100110 16 × 26 000000000100011 32 × 35 ….000000000101110111 375 = 11(mod 13)

Fig. (b) Fuzzy Modular Multiplication Fig. 2. Numerical Example of Modular Multiplication with and without Fuzzy

+

c

+ R'n

c

R'0

c

b0b1bn

mtn mt1 mt0

c

+ +R'1

Fuzzy Modular Multiplication

R = A ⋅ B (mod m) R′ = A ⋅ B – z m

Fig. 3. Simplified Hardware Architecture for Fuzzy Modular Multiplication

a0 a1 a2 ...AA Serial

=

t.mb0 b1 b2 ...B

Multiplier=


ISBN:978-988-17012-6-8 WCECS 2009

V. IMPLEMENTATION OF ECC USING FUZZY MODULAR ARITHMETIC

Algorithm 2 EC point addition and doubling using Fuzzy Modular Arithmetic Input : p1= (x1,y1), p2 = (x2,y2) and m Output : p1 + p2 = p3 = (x3,y3) 1.

1λT ← y2 – y1

2. 2λT ← x2 – x1

3. m2λT ← 2m

λ2T − mod m (Fermat’s Theorem)

4. 1λm2

T− ← mzT 1λm2− ( z1m ≤

m2λT )

5. λ←1λT 1

λm2T−

6. 3xT ← λ2 – x1 – x2

7. If 3xT > m then

8. x3 ← mzT 2x 3− (

3x2 Tmz ≤ )

else 9. x3 ← 3xT

10. end if 11.

3yT ← (x1 – x3) λ – y1

12. If 3yT > m then

13. y3 ← mzT 3y3−

else 14. y3 ← 3yT

15. end if

Input : p1= (x1,y1), a and m Output : 2 p1 = p3 = (x3,y3) 1.

1λT ← a x3 21 +

2. 2λT ← 2 y1

3. m2λT ← 2m

λ2T − mod m (Fermat’s Theorem)

4. 1λm2

T− ← mzT 1λm2− ( z1 m ≤

m2λT )

5. λ←1λT 1

λm2T−

6. 3xT ← λ2 – 2 x1

7. If 3xT > m then

8. x3 ← mzT 2x3− (

m2λ2 Tmz ≤ )

else 9. x3 ← 3xT

10. end if 11.

3yT ← (x1 – x3) λ – y1

12. If 3yT > m then

13. y3 ← m zT 3y3−

else 14. y3 ← 3yT

15. end if

The execution of ECC scheme is mostly dominated by the efficient implementation of finite field arithmetic and point multiplication kP. To compute kP, several methods such as binary method, binary NAF method, and windowed NAF. have been proposed [10]. To implement these methods, efficient hardwired architecture and a platform that support the efficient computation of such methods are needed. Previous hardware work includes: the first ASIC implementation with Motorola M68008 microcomputer, reconfigurable finite-field multiplier, parallelized field multiplier [7], [18]. The main objective of this work is to perform modular operations without trail division because the hardware implementation of trial division is an expensive. Thus, this work emphasizes fuzzy modular arithmetic with no division and division is repeatedly replaced by subtraction and hence the hardware implementation is relatively easy. Moreover, to perform kP, repeated addition is used. To add two points of EC, in both formula modular multiplication inversions is involved and it takes considerable amount of time. To avoid it, Fermat’s Theorem is used in this paper which is illustrated in section A.2 .The EC point addition and doubling using Fuzzy Modular Arithmetic and Fermat’s theorem is given in Algorithm 2. Moreover, this work further focuses on the implementation of ECC with AT89C51 microcontroller in which the simplified fuzzy modular multiplier is embedded. AT89C51 is a low-power; high performance CMOS 8-bit microcomputer with 4K bytes of Flash programmable and erasable read only memory. The on-chip Flash allows the program memory to be reprogrammed in-system.

VI. EXPERIMENTAL RESULT This section summarizes the performance of the hardware implementation of ECC with and without fuzzy modular arithmetic. The software implementation of the work is performed in Visual C++ version 6.0 IDE, Advanced C version 4.0. The time taken for encryption and decryption operations of ECC for the various file sized message with and without fuzzy modular arithmetic is shown in Table I. Table I Encryption and Decryption time of ECC with and without Fuzzy Modular Arithmetic

ECC without Fuzzy modular Arithmetic

(Time in ms)

ECC with Fuzzy modular Arithmetic

(Time in ms)

File size (KB)

E D T E D T 1 3 5 7 8

1490 2470 3510 4573 5685

1370 2410 3460 4512 5642

2860 4880 6970 9085

11327

1320 1920 2820 3916 4613

1340 1980 2890 3996 4663

2660 3900 5710 7912 9276

KB- Kilo Bytes, ms- mille seconds, E- Encryption Time, D-Decryption Time, and T- Total Time From Table I, it is observed that the time taken for encryption and decryption with fuzzy modular arithmetic is substantially decreased compared to corresponding counterpart without fuzzy modular arithmetic. Further, it is observed that encryption time is moderately larger than decryption time in both cases.


ISBN:978-988-17012-6-8 WCECS 2009

VII. CONCLUSION Microcontroller based implementation of ECC is thought of and implemented. To increase the speed of encryption and decryption operations fuzzy modular arithmetic and Fermat’s theorem are used. In fuzzy modular arithmetic to perform modulus operation, repeated subtraction is used instead of division and hence hardware complexity is relatively reduced than division operation. To perform inversion operation with respect to modulus, Fermat’s theorem is used which is further simplified by repeated multiplication.

REFERENCES

[1] W. Adi, “Fuzzy Modular Arithmetic for Cryptographic Schemes with Applications for Mobile Security”, IEEE, 2000, pp. 263-265.

[2] S. B. Ors, L. Batina, B. Preneael, and J. Vandewalle, “Hardware Implementation of an Ellipic Curve Processor over GF (p)”, in Proceedings of the IEEE Conference on Application Specific Systems users and Processors (ASSP), 2000, pp. 433-443.

[3] R. Cheung, W. Telle, and P. Cheung, “Customizable Elliptic Curve Cryptosystems”, IEEE, TVLSI, Vol. 13(a), September, 2005 , pp.1048-1049.

[4] D.Hankerson, L. Herandez, and A. Menezes, “Software Implementation of Elliptic Curve Cryptography over Binary Fields”, CHES, LNCS 1965, Berlin Heidelberg, Springer-Verlag, 2000, pp. 1-24.

[5] A. Hanoun, W. Adi , F. Mayer-Lindenberg, and B.Soundan , “Fuzzy Modular Multiplication Architecture and Low Complexity IPR-Protection for FPGA Technology”, IEEE, 2006, pp. 325-326.

[6] D. Knuth, “The Art of Computer Programming – Semi Numerical Algorithms”, Vol.2, Addison-Wesley, Third Edition, 1998.

[7] J. Lutz, and A. Hasan, “High Performance FPGA based Elliptic Curve Cryptographic Co-processor”, In ITCC 04, International Conference on Information Technology Coding and Computing, Vol. 2, 2004, pp. 486.

[8] J. Julio Lopez, and R. Dahab, “An Overview of Elliptic Curve Cryptography”, Reconfigurable Architecture Workshop (RAW), Nice, France, April 22, 2003, pp. 1-28.

[9] N. Koblitz, “A Course in Number Theory and Cryptography”, Second Edition, Springer-Verlag, 1994.

[10] N. Koblitz, “Elliptic Curve Cryptosystems”, Mathematics of Computation Vol. 48 (177), November, 1982, pp. 203-209.

[11] A. Menezes, and S.Vanstone, “Elliptic Curve Cryptosystems and their implementation”, Journal of Cryptology 6, 1993, pp. 209-224.

[12] G. Orlando, and C. Paar, “A High Performance Reconfigurable Elliptic Curve Processor for GF(2m)”, CHES, Lectures Notes in Computer Science 1965, Springer-Verlag, 2000, pp. 41-56.

[13] M. Morales, Sandoval, and Claudia Feregrino-Uribe, “GF(2m) Arithmetic Modulus for Elliptic Curve Cryptography”, IEEE, 2006.

[14] A. Menezes, P. Van Oorschot, and S. Vanstone, “Handbook of Applied Cryptography”, CRC Press, 1997.

[15] I. Blake, G. Seroussi, and N.P. Smart, “Elliptic Curves in Cryptography”, Ser. London Math. Soc. Lecture Note Series, Cambridge Univ. Press, 1999.

[16] S. B. Ors, L. B. Batina , and J. Vandewalle, “Hardware Implementation of a Montegomery Modular Multiplier in a systolic array”, In the 10th Reconfigurable Architecture Workshop (RAW), Nice, France, 2003.

[17] T. Wollinger, J. Guajardo, and C. Christof Paar, “Cryptography in Embedded Systems: An Overview”, Proceedings of the Embedded World 2003 Exhibition and Conference, Design Electronik, Nuernberg, Germany, February 18-20, 2003, pp. 735-772.

[18] Yong-Je Choi, Moo-Seop Kim, Hang-Rok Lee, and Ho-Wan Kim , “Implementation and analysis of Elliptic Curve Cryptosystems over Polynomial basis and ONB”, PWAEST, Vol. 10, December, 2005, pp. 130-134.


ISBN:978-988-17012-6-8 WCECS 2009

A.4

Chapter

1

INFORMATION SECURITY ENHANCEMENT TO PUBLIC –KEY

CRYPTOSYSTEM THROUGH MAGIC SQUARES

Author Gopinath Ganapathy, and K. Mani

Dr. Gopinath Ganapathy, Professor and Head with the computer Science Department, University of

Bharthidasan, Trichy, India – 620 024. (Phone : +91 9842407008; email: [email protected]).

K. Mani is with the computer Science Department of Nehru Memorial College, Puthanapatti,

University of Bharthidasan, Trichy, India – 621 007. (Phone : +91 9443598804; email:

[email protected]).

Abstract The efficiency of a cryptographic algorithm is based on its time taken for encryption

or decryption and the way it produces different cipher text from a plain text. The RSA,

the widely used public key algorithm and other public key algorithms may not

guarantee that the cipher text is fully secured. As an alternative approach to handling

ASCII value of the text in the cryptosystems, a magic square of order 16 equivalent to

ASCII set generated from two different approaches is thought of in this work. It attempts to enhance the efficiency by providing add-on security to the cryptosystem.

This approach will increase the security due to its complexity in encryption because it

deals with the magic square formation with seed number, start number and sum that

cannot be easily traced. Here, encryption / decryption is based on numerals generated

by magic square rather than ASCII values. This proposed work provides another layer of security to any public key algorithms such as RSA, ElGamal etc., Since, this model is

acting as a wrapper to a public key algorithm, it ensures that the security is enhanced.

Further, this approach is experimented in a simulated environment with 2, 4, and 8

processor model using Maui scheduler which is based on back filling philosophy.

(i) Paper Published in International Journal – Springer Verlag

(Gopinath Ganapathy and K. Mani, “Information Security Enhancement of Public-key Cryptosystem Through Magic Squares”, Machine Learning and System Engineering, Lecture Notes in Electrical Engineering, Vol. 68, Springer, pp. 423-437.)

mailto:[email protected]

mailto:[email protected]

2 Chapter

Index Terms —Magic Square, Public-Key Cryptosystem, Fundamental Magic

Square, Magic Square based on properties, RSA, and Security.

INTRODUCTION

Today security is an important thing that we need to send data from one location

to another safely. As there is strong security, there is a great hacker and spire at the

other side. Therefore, many models and systems are looking for optimizat ion of the

electronic connected-world. Cryptography accepts the challenges and plays the vital

role of the modern secure communication world. It is the study of mathemat ical

techniques related to aspects of information security such as confidentiality, data

integrity, entity authentication, and data origin authentication. Cryptographic

algorithms are divided into public-key and secret-key algorithms. In public-key

algorithms both public and private keys are used, with the private key computed

from the public key. Sec ret-key algorithms rely on secure distribution and

management of the session key, which is used for encrypting and decrypting all

messages. Though, public-key encryption is slower than symmetric -key encryption,

it is used for bulk-data encryption. Hence, in practice cryptosystems are a mixture of

both[1], [2].

There are two basic approaches used to speed up the cryptographic

transformations. The first approach is to design faster (symmetric or asymmetric)

cryptographic algorithms. Th is approach is not available most of the time. The speed

of cryptographic algorithm is typically determined by the number of rounds (in

private-key) or by the size of messages (in public-key case). The second approach is

the parallel cryptographic system. The main idea is to take a large message block,

divide it into blocks of equal sizes and each block can be assigned to one processor

[3]. To perform the operations in parallel, effect ive scheduling is very important so

that it can reduce the waiting time of message for processing. The back filling is one

such approach.

The security of many cryptographic systems depends upon the generation of

unpredictable components such as the key stream in the one-time pad, the secret key

in the DES algorithms, the prime p, and q in the RSA encryption etc. In all these

cases, the quantities generated must be sufficient in size and the random in the sense

that the probability of any particular value being selected must be sufficiently small.

However, RSA is not semantically secure or secure against chosen cipher text

attacks even if all parameters are chosen in such a way that it is in feasible to

compute the secret key d from the public key (n, e), choosing p, q are very large etc.

Even if the above said parameters are taken carefu lly, none of the computational

problems are fu lly secured enough [4]. Because to encrypt the plaintext characters,

their ASCII values are taken and if a character occurs in several places in a plaintext

there is a possibility of same the cipher text is produced. To overcome the problem,

this paper attempts to develop a method with different doubly even magic squares of

order 16 by two different approaches and each magic square is considered as one

ASCII table. Thus, instead of taking ASCII values for the characters to encrypt,

preferably d ifferent numerals representing the position of ASCII values are taken

from magic square and encryption is performed using RSA cryptosystem.

METHODOLOGY

. 3

Cipher Text Encryption Algorithm

TextClear theof value

ASCII

Sum Square

Magic

and

No., Seed

No., Starting

Generate Doubly Even Even Magic

Square

Clear Text

Determine the numeral value from the Magic Square

that corresponds to the position of

ASCII value

Numeral

ASCII

value of the

Clear text

Determine the position

of the numeral

value in the Magic

Square

Cipher Text Decryption Algorithm

NumeralClear Text

The working methodology of the proposed add-on security model is discussed

stepwise.

Construct different doubly even magic square of order 16 as far as possible and

each magic square corresponds to one ASCII set .

(a) Encryption process

(b) Decryption process

Figure #-1. Add-On Wrapper Model

To encrypt the character, use the ASCII value of the character to determine the

numeral in the magic square by considering the position in it. Let NP and NC

denote the numeral of the plaintext and cipher text respectively. Based on NP

and NC values, all p laintext and cipher text characters are encrypted and

decrypted respectively using RSA algorithm.

To speed up the operations, perform them in parallel in a simulated environment.

For that, use Maui scheduler with back filling philosophy.

The methodology of Add-on Security Model is shown in Figure #- 1

1.1 Magic Squares and their Construction

Definition 1: A magic square of order n is an arrangement of integers in an n*n

magic square such that the sums of all the elements in every row, column and along

the two main diagonals are equal.

A normal magic square contains the integers from 1 to n2. It exists for all

orders n 1 except n = 2, although the case n = 1 is trivial as it consists of a single

4 Chapter

cell containing the number. The constant sum in every row, co lumn and diagonal is

called the magic constant or magic sum, M[5]. The magic constant of a normal

magic square depends only on n and has the value

2

1)2

n(nM(n)

(1)

For normal magic squares of order n = 3, 4, 5, 6, 7, 8 … the magic constants are

15, 34, 65, 111, 175, 265… respectively. Though there is only one magic square of

order 3 apart from trivial notations and reflections, the number of squares per order

quickly sky rockets. There are 880 distinct order 4 squares, and 275,305,234 distinct

order 5 magic squares [6]. Magic squares can be classified into three types: odd,

doubly even (n divisible by four) and singly even [7] (n even but not divisible by

four). This paper focuses only on doubly even magic square implementation and

their usefulness for public-key cryptosystem

While constructing the doubly even magic squares of order 16 based on different

views and fundamental properties of magic square the following notations are used.

MS : Magic Square

n : Order of MS where n = 4m, where

m = 1, 2, 3 and 4

MSn : MS of order n

MSB4 : Base MS o f order 4

MSstart : Starting number of MS

MSTnsum : Total sum of MS of order n

MSDn sum : Diagonal sum of MS o f order n

T_No_MS : Total Number of MS

S1, S2 S3, S4 : First, Second, Third and Fourth digit of 4 digits seed number.

Each digit has a value from 0 to 7. If S1 = S2 = S3 = S4 = 0, i.e 0000 then it is

MSB4. The values in the MSB4 are filled as shown in Figure #-2. Call it as

MS4_fill_order (Min start , Max start)

-4 MS start -8 +12

-10 +14 -6 +2

+8 -12 +4 MST2sum

+6 -2 +10 -14

Figure # -2. Magic Square Filling Order

where the numerals with positive sign represent value to be added with MS start

and the numerals with negative sign represent value to be subtracted from

MST2sum..

MS4_min_start_value : Minimum starting value of MS

MS4_fill_order (MSstart, MST4sum) : function used to fill the values in MS the

order shown in Figure # -2

MSn_base i : ith nth

order base MS

. 5

MS4_sub i : ith 4th

order MS obtained from MSB4

by using suitable transformation based

on S i , where i= 1,2,3,and 4

MS4_prop_sub i : ith 4th

order property based MS

obtained from MSB4 by using the

fundamental property of MS4

magic square based on Sp i ,

where p i = 1,2,3,and 4

Prop : properties of fundamentalMS4

with four character values

|| :concatenation

1.2 Construction of Doubly Even Magic Square based

on different views of fundamental Magic Square

In this work, to generate the doubly even magic square, any seed number, starting

number, and magic sum may be used and the numbers generated will not be in

consecutive order. The algorithm 1.2.1 proposed by Gopinath Ganapathy, and Mani

[8] starts with building 4 x 4 magic square. Incrementally 8 x 8 and 16 x 16 magic

squares are built using 4 x 4 magic squares as building blocks.

1.2.1 Algorithm

Input : 4 digit Seed number, Starting Number and

Magic Square Sum

Output: Doubly Even Magic Square of order 16.

1. Read MSstart ; Si, i = 1,2,3,4 ;MST16sum and T_No_MS

2. MST4sum MST16sum / 4 ; 3. MST8sum MST16sum / 2

4. Count 1

5. While Count T_No_MS do // to generate MS16

5.a. prev 1;

5. b. for j 1 to 4 // to generate four MS8

5.b.1 for i 1 to 4 // to generate four MS4

5.b.1.1 if ((j=2)&&(i = 3)) || ((j = 3) &&(i =2))

MS4_min_start_value

MS4start + 16 *( iprev-1) +1

else

MS4_min_start_value

MSstart + 16* (prev-1 )

end if

5.b.1.2 MS4_max_start_value

MST4sum – 16* (prev- 1) - MSstart

5.b.1.3 MS4_base i call MS4_fill_order

(MS4_min_start_value, MS4_max_ start_value)

6 Chapter

5.b.1.4 Case Si in 0,1,2,3,4,5,6,7

0: MS4_sub i rotate MS4_base i by 0

1: MS4_sub i rotate MS4 _base i by 90



4: MS4_sub i rotate MS4_base i by 180 about a

vertical axis

5: MS4_sub i rotate MS4_base i by 90 through

anticlock wise

6: MS4_sub i rotate MS4_base i by 180 about

horizontal axis

7: MS4_sub i rotate MS4_base i by 180 about the

main diagonal

end case Si

5.b.1.5 prev prev+1

end for i

5.b.2 MS8_sub j MS4_sub 1|| MS4_sub 2 || MS4_sub 3 || MS4_sub 4

5.b.3 case j in 2,3,4

2: S1 S2 ; S2 S3 ; S3 S4 ; S4 S1

3: S1 S3 ; S2 S4; S3 S1 ; S4 S2

4: S1 S4 ; S2 S1 ; S3 S2 ; S4 S3

end case j

5.b.4 Si S1 || S2 || S3 || S4 , i 1,2,3,4

end for j

5.c MS16_count MS8_sub1|| MS8_sub2 ||MS8_sub3||MS8_sub4

5.d Si (Si + rnd(10)) mod 8 , i = 1,2,3,4

5.e count count + 1

end while count

1.2.2 Example

Based on the Algorithm 1.2.1 the first four MS4_sub i , i=1,2,3 and 4 are

generated and they are shown in Figure #-3,4,5, and 6.

1535 4 1531 16

1529 18 1533 6

12 1527 8 1539

10 1537 14 1525

Figure #-3. MS4_sub1 with (4,1539)

. 7

Figure (a). MS4_sub2 with (20, 1523) Figure( b) . 90° rotation of Figuare(a)

Figure #-4. MS4_sub2 with (20, 1523)

Figure #-5. MS4_sub3 with (36, 1507) Figure #-6. MS4_sub4 with (52, 1491)

Other magic squares MS8_sub2, MS8_sub3 and MS4_sub4 are generated in this

manner and they are concatenated so that they form MS16 which is shown in Figure

#-7.

1535 4 1531 16 26 28 1513 1519 74 76 1465 1471 1455 85 1451 96

1529 18 1533 6 1521 1511 34 20 1473 1463 82 68 1449 98 1454 86

12 1527 8 1539 30 24 1517 1515 78 72 1469 1467 93 1447 88 1459

10 1537 14 1525 1509 1523 22 32 1461 1475 70 80 90 1457 94 1446

1503 36 1499 48 1487 52 1483 64 1439 101 1435 112 1423 116 1419 128

1497 50 1501 38 1481 66 1485 54 1433 114 1438 102 1417 130 1421 118

44 1495 40 1507 60 1479 56 1491 109 1431 104 1443 124 1415 120 1427

42 1505 46 1493 58 1489 62 1477 106 1441 110 1430 122 1425 126 1413

1407 132 1403 144 1391 149 1387 160 1343 196 1339 208 1327 212 1323 224

1401 146 1405 134 1385 162 1390 150 1337 210 1341 198 1321 226 1325 214

140 1399 136 1411 157 1383 152 1395 204 1335 200 1347 220 1319 216 1331

138 1409 142 1397 154 1393 158 1382 202 1345 206 1333 218 1329 222 1317

1375 165 1371 176 186 188 1353 1359 1311 228 1307 240 250 252 1289 1295

1369 178 1374 166 1361 1351 194 180 1305 242 1309 230 1297 1287 258 244

173 1367 168 1379 190 184 1357 1355 236 1303 232 1315 254 248 1293 1291

170 1377 174 1366 1349 1363 182 192 234 1313 238 1301 1285 1299 246 256

Figure #-7. First MS16 using Starting No. 4, Seed No. 0100, and Magic Sum 12345

26 28 1513 1519

1521 1511 34 20

30 24 1517 1515

1509 1523 22 32

1519 20 1515 32

1513 34 1517 22

28 1511 24 1523

26 1521 30 1509

1487 52 1483 64

1481 66 1485 54

60 1479 56 1491

58 1489 62 1477

1503 36 1499 48

1497 50 1501 38

44 1495 40 1507

42 1505 46 1493

8 Chapter

Similarly other MS16 magic squares are generated by using suitable

transformations of the seed number. In this paper, only four MS16 magic squares are

generated.

1.3 Construction of Doubly Even Magic Square of order

16 based on the properties of 44 Magic Square

In this paper, section 1.2 focuses on construction of a 1616 magic square

from sixteen 44 different views of fundamental magic squares. To obtain the

different views of magic squares, each one can be rotated or reflected so that it can

generate eight different views of a 44 fundamental magic square. Thus, any square

is one of a set of eight that are equivalent under rotation or reflection. In order to

generate different views of a magic square, the general format shown in Figure # -8

is used.

A B C D

E F G H

I J K L

M N O P

Figure # -8. General format of a fundamental 4×4 magic square

1.3.1 Sub-types of 4x4 Magic Square

Normal

A magic square is said to be normal in which every row, column, and

the main and secondary diagonals sum to the magic sum. That is B+C+N+O =

E+H+I+L = F+G+J+K = A+D+M+P = MST4sum

Pan–Magic

It has the additional property that the broken diagonals also sum to the

magic sum. That is B+G+L+M = D+E+J+O = C+F+I+P = A+H+K+N = C+H+I+N

= B+E+L+O = MST4sum .The magic squares depicted in Figure #- 3, 4, 5, and 6 are

normal and Pan Magic.

Associative

A magic square is an associative, it has the additional property that the

eight symmetrically opposite the centre of the square shown in Figure# -8 are AP,

BO, CN, DM, EL, FK, GJ and HI having the magic sum magic constant MST4sum.

Quadrant Associative

It has the property that all diagonally opposite pairs of numbers within

the four main quadrants sum to MST4sum. That is, the diagonally opposite pairs of

numbers within the four quadrants AF, BE, CH, DG, IN, JM, KP and LO are having

the magic sum MST4sum .

. 9

Distributive

It has the property that each of the four integers in the set of numbers

(1,2,3,4), and (5,6,7,8) and (9,10,11,12), and (13,14,15,16) are located in a row and

column where none of the other three numbers in the set are located [9], [10].

1.3.2 Construction of Doubly Even Magic Square of order 16 based

on the properties

In this work, to generate the doubly even magic square based on the properties the

algorithm 1.2.3 is used similar to algorithm 1.1.1 with a slight modification. Apart

from any starting number, and magic sum, the fundamental mag ic square seed

number 0000, and the four character value for the Prop variab le are g iven as input.

The algorithm starts with by constructing 4 x 4 fundamental magic square and there

are sixteen 4 x 4 fundamental magic squares are generated. In each magic square,

the properties of fundamental magic square are applied based on the cyclic left shift

operation in the value of Prop variable. The algorithm is shown in 1.3.3.

1.3.3 Algorithm

Input : 4 digit seed number, starting Number, magic square property , and

magic squares sum

Output: Doubly Even Magic Square of order 16.

To generate 4 x 4 fundamental magic squares, the algorithm uses the steps from

1 to 5.b.1.3 illustrated in Algorithm 1.2.1. To generate further, the following steps

are used. 5.b.1.4 Case Sp i in P. A, Q, D

‘P’: MS4_prop_sub i Pan-magic of MS4_base i

‘A’: MS4_prop_sub i Associativec of MS4_base i

‘Q’: MS4_prop_sub i Quadrant Associative of MS4_base i

‘D’: MS4_prop_sub i Distributive of MS4_base i

end Case

5.b.1.5 prev prev+1 end for i

5.b.2 MS8_prop_sub j MS4_prop_sub 1|| MS4_prop_sub 2

|| MS4_prop_sub 3 || MS4_prop_sub 4 5.b.3 case j in 2,3,4

2: Sp1 Sp2 ; Sp2 Sp3; Sp3 Sp4 ; Sp4 Sp1



end case j

5.b.4 Spi Sp1 || Sp2 || Sp3 || Sp4 , i 1,2,3,4 end for j

5.c MSP16_count

MS8_prop_sub1||MS8_prop_sub2||MS8_prop_sub3||MS8_prop_sub4

5.d Si (Si + rnd(10)) mod 8 , i = 1,2,3,4

10 Chapter

5.e count count + 1 end while count

1.3.4 Magic Square Construction based on properties-

Example

To generate a 1616 doubly even magic square equivalent to ASCII set based on

the properties of 44 magic square, already generated the fundamental magic

squares MS4_sub1, MS4_sub2, MS4_sub3 and MS4_sub4 are considered. Let , the

value for seed number , starting number , magic sum and Prop are given 0000, 4,

12345 and ‗PAQD‘ respectively as input. The Figure # -3, -4a, -5, and 6 are

considered and the 44 magic squares based on the properties are generated as

follows.

Since the first value of ‗Prop‘ variab le is P i.e pan-magic, apply this property

to the MS4_sub1. The resulting MS4_prop_sub1 is shown in Figure #-9.Based on

the second value ‗A‘ i.e associative, the MS4_prop_sub2 is generated from

MS4_sub2 and it is shown in Figure# -10.

Figure #-9. MS4_prop_sub1 with (4, 1539) Figure #-10. MS4_prop_sub2 with (20, 1523)

Since the third character is ‗Q‘ i.e quadrant associative, the

MS4_prop_sub3 is generated from MS4_sub3 and it is shown in Figure # -11.

Figure #-11. MS4_sub3 with (36, 1507) Figure #-12. MS4_sub4 with (52, 1491)

Now, the fourth letter is ‗D‘ i.e distributive the MS4_prop_sub4 is

generated from MS4_sub4 and it is shown in Figure# -12.

Thus, to form MS8_prop_sub1 concatenate all four MS4_prop_subi , i=1,2,3 and

4 (shown in Figures #-9,10,11,and 12) and the first quadrant 88 magic square is

shown in Figure #-13.

1523 1503 22 32

428 26 1513 1519

24 30 1517 1515

1511 1521 34 20

1535 4 1531 16

1529 18 1533 6

12 1527 8 1539

10 1537 14 1525

52 1489 62 1483

66 1479 56 1485

1481 60 1491 54

1487 58 1477 64

1507 42 1493 44

1501 36 1499 50

38 1503 48 1497

40 1505 40 1495

. 11

MS4_prop_sub1 MS4_prop_sub2


Figure #-13. MS8_prop_sub1 with (4,1539)

The second quadrant MS8_prop_sub2 magic square is generated by

transforming the Seed No. as Sp1 Sp2 ; Sp2 Sp3; Sp3 Sp4 ; Sp4 Sp1and the

‗Prop‘ value is changed to ‗AQDP‘, the second quadrant 88 magic square is

generated from the next four fundamental 44 magic squares shown in Figure #-14.

1471 68 1467 80 1456 84 1451 96

1465 82 1469 70 1449 98 1454 86

76 1463 72 1475 92 1448 88 1459

74 1473 78 1461 90 1457 94 1446

1439 101 1435 112 1423 116 1419 128

1433 114 1438 102 1417 130 1421 118

109 1431 104 1443 124 1415 120 1427

106 1441 110 1430 122 1425 126 1413

Figure # -14. MS8_prop_sub2 with (68,1475)

1411 138 1397 140 149 1393 158 1387

1405 132 1403 146 162 1383 152 1390

134 1407 144 1401 1385 157 1395 150

136 1409 142 1399 1391 154 1382 160

1375 165 1371 176 1363 1349 182 192

1369 178 1374 166 188 186 1353 1359

173 1367 168 1379 184 190 1357 1355

170 1377 174 1366 1351 1361 194 180

Figure # -15. MS8_prop_sub3 with (132, 1411)

The third quadrant MS8_prop_sub3 is obtained by transforming the seed number

as Sp1 Sp3 ;Sp2 Sp4; Sp3 Sp1 ; Sp4 Sp2 and the ‗Prop‘ value is changed

to ‗QDPA‘. By applying the properties , the resultant magic square is shown is

Figure -15.

Similarly, the fourth quadrant MS8_prop_sub4 is generated in this manner

by shifting ‗QDPA‘ one position towards left, the resultant magic square is shown in

Figure #-16.

12 Chapter

196 1345 206 1339 1327 212 1323 224

210 1335 200 1341 1321 226 1325 214

1337 204 1347 198 220 1319 216 1331

1343 202 1333 208 218 1329 222 1317

1315 1301 230 240 1291 1285 256 254

236 234 1305 1311 258 252 1289 1287

232 238 1309 1307 244 250 1295 1297

1303 1313 242 228 1293 1299 246 248

Figure #-16. MS8_prop_sub4 with (196, 1347)

In order to obtain first MS16_prop magic square, concatenate all four

MS8_prop_subi ( i = 1,2,3,and 4 ) magic squares and it is shown in Figure# -17.



Figure #-17. First MS16_prop Magic Square using Starting No. 4, Seed No. 0100, ‗PAQD‘,

and Magic Sum 12345

Similarly, the other MS16_prop magic squares are generated by using suitable

transformations of the seed number.

1.4 Encryption / Decryption of Plain Text using RSA

Cryptosystem with Magic Square

To show the relevance of this work to the security of public-key encryption

schemes, a public-key cryptosystem RSA is taken. RSA was proposed by Rivest et

al. To encrypt a plaintext M the user computes C = Me mod n and decryption is done

by calculating M = Cd mod n. In order to thwart currently known attacks, the

modulus n and thus M and C should have a length of 512-1024 bits in this paper .

1.4.1 Wrapper implementation-Example

In order to get a proper understanding of the subject matter of this paper, let p =

11, q = 17 and e = 7, then n = 11(17) = 187, (p-1)(q-1) = 10(16) = 160. Now d =

23. To encrypt, C = M7 mod 187 and to decrypt M = C

23 mod 187. Suppose the

message is to be encrypted is ―BABA‖. The ASCII values for A and B are 65 and 66

respectively. To encrypt B, the numerals which occur at 66th

position in first (Figure

#-7) and third MS16(not shown here) are taken because B occurs in first and third

. 13

position in the clear text. Similarly, to encrypt A, the numerals at 65th

position in

second and fourth MS16(not shown here) are taken. Thus NP(A) = 42 and 48, NP(B)

= 36 and 44. Hence NC(B) = 367mod 187 = 9, NC (A) = 42

7 mod 187 = 15, NC (B) =

447 mod 187 = 22, NC (A) = 48

7 mod 187 = 157. Thus, for same A and B which

occur more than once, different cipher texts are produced. Similar encryption and

decryption operations can be performed by cosidering the numerals occur at the

positions of 66 and 65 for the characters B and A respectively from MS16_prop

magic square .

1.5 Parallel Cryptography

To speed up cryptographic transformation, the parallel cryptography is used.

Effective scheduling is important to improve the performance of crypto system in

parallel. The scheduling algorithms are divided into two classes: time sharing and

space sharing. Backfilling is the space sharing optimizat ion technique.Maui is a job

scheduler specifically designed to optimize system utilization in po licy driver,

heterogeneous high performance cluster (HPC) environment. The philosophy behind

it is essentially schedule jobs in priority order, and then backfill in the holes . Maui

has a two-phase scheduling algorithm. In the first phase, the high priority jobs are

scheduled using advanced reservation. A backfill algorithm is used in the second

phase to schedule low-priority jobs between previously selected jobs [11]. In our

study the Maui Scheduler with back filling scheduling technique is used to calculate

the encryption/decryption time of g iven message in simulated environment.

EXPERIMENTAL RESULT

The methodology proposed is implemented in visual C++ version 6.0. The time

taken for encryption and decryption of various file sized message in simulated

parallel environment using RSA public-key crypto system based on the properties

of magic squares is shown in Figure #-18. The simulation scenario configured in

our implementation consisting of 2, 4, and 8 processors.

From Figure #-18 we observe that as the file size is increased in double,

encryption and decryption time is also increased in double for single processor.

Moreover, the time taken for encryption and decryption is almost same. Parallel

encryption and decryption have more effect if the file size is increased.

File

Size (MB)

No.of Processors

1 2 4 8

E ms

D ms

T ms

E ms

D ms

T ms

E ms

D ms

T ms

E ms

D ms

T ms

1 362 405 767 560 607 1167 502 515 1017 450 447 897

2 700 707 1407 737 756 1493 590 599 1189 514 530 1044

3 1332 1370 2702 1073 1097 2170 753 672 1425 610 605 1215

4 2601 2618 5219 1700 1741 3441 1084 1100 2184 765 776 1541

E.Encryption time, D.Decryption time, and ms-milliseconds

Figure # -19 . Encryption and decryption time using RSA on a Pentium Processor

Based on Properties of Magic Squares

14 Chapter

CONCLUSION

An alternative approach to existing ASCII based cryptosystem a number based

on magic squares generated by two different schemes is thought of and

implemented. It is noted that that the numerals generated using different views of

magic square for encryption may be obtained by the eavesdropper(but it is very

difficult) because of some kind of relat ionship exists among the numerals. But, in

the case of magic squares generated using the properties, it is very difficult to

obtain the numerals by him because of the numbers do not have any kind of

relationship and hence it is very difficult to obtain such numerals for encryption.

Thus, the second approach is better than the first one. This methodology will add -

on one more layer of security, it adds numerals for the text even before feed ing into

public-key algorithm for encryption. Th is add-on security is built using magic

square position equivalent to the character to be encrypted.

Further efficiency of cryptographic operation depends on performing them in

parallel. Simulation using different number of processors for encryption /decryption

has shown that the time taken for decryption is approximately 1.2 t imes larger than

the corresponding time for encryption.

References

1. Antti Hamalainenn, Matti Tommiska, and Jorma Skytta, 6.78 Gigabits per Second Implementation of

the IDEA Cryptographic Algorithm LNCS 2438, Springer-Verlag, 2002, p. 760-769. 2. Thomas Wollinger, Jorge Guajardo, and Chirst of Paar,Cryptography in Embedded Systems: An

overview. In Proc. of the Embedded World 2003 Exhibition and Conference, Feb. 18-20, 2003, p. 735-744.

3. Josef Piepryk, and David Pointcheval, Parallel Authentication and Public-Key Encryption, The Eigth Australasian Conference on Information Security and Privacy, Springer-Verlag, Jul. 2003, p.

383-401. 4. A. J. Menezes, P.C. Van Oorschot, and S. Vanstone, Handbook of Applied Cryptography, (CRC

Press, Boca Ration, Florida, USA, 1997).

5. http://en.wikipedia.org/wiki/Magic_ squares , p. 1-3. 6. Adam Rogers, and Peter Loly, The Inertial Properties of Magic Squares and Cube, Nov. 2004, p. 1-3. 7. Clifford A. Pickover, The Zen of Magic Squares, Circles and Stars (Universities Press, Hyderabad,

India, 2002). 8. Gopinath Ganapathy, and K. Mani, Add-On Security Model for Public-KeyCryptosystem Based on

Magic Square Implementation, WCECS, (October 20-22, 2009, San Francisco, USA), p. 317-321. 9. Gary A Foster, St.Joseph‘s Healthcare Hamilton, Hamilton, et al, (Global Forum 2009) SAS ® Magic

Squares. 10. John P.Robertson ,Theory of 4 × 4 Magic Squares (January 17, 2009). 11. Y.Zhang, H. Franke, J. E. Moreira and A. Sivasubramanian, An Integrated Approach to Parallel

Scheduling using Gang-Scheduling, Backfilling andMigration, JSSPP, UK, Springer-Verlag, 2006, p. 133-158.

http://en.wikipedia.org/wiki/Magic_%20squares

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