NCM Latin squares talk

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Combinatorial conditions for secret sharingfor Public Key Cryptography

byDr. N. Chandramowliswaran

ProfessorApplied Sciences, NCU, Gurgaon

FEB. 27, 2016

Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 1 / 19

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Latin Squares

Definition 1Let n be a given positive integer and S = {s1, s2, . . . , sn} be a given setof n distinct elements.

A Latin square of order n based on S is an n× n array in which everyelement of S occurs exactly once in each row and exactly once in eachcolumn.

Thus each of the rows and columns of a Latin square is a permutationof S.


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Let G be a finite group of n elements

Suppose f : G×G → G such that f(gi, gj) = gi ∗ gj ∈ G

For a fixed i, gi ∗ gj is a permutation on G for all j varies from 1 to n

Similarly, for a fixed j, gj ∗ gi is a permutation on G for all i variesfrom 1 to n

such a function f is called a Latin function defined on G×G to G


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Let G = {g1, g2, . . . , gi, . . . , gj , . . . , gn}

Assume g1 = e, the identity element of G

Define an n× n matrix A by A = [aij ] = [gi ∗ gj ]

A is a Latin square of order n based on G

the i-th row ofA = {gi ∗ g1, gi ∗ g2, . . . , gi ∗ gi, . . . , gi ∗ gj , . . . , gi ∗ gn} andthe j-th column ofA = {g1 ∗ gj , g2 ∗ gj , . . . , gi ∗ gj , . . . , gj ∗ gj , . . . , gn ∗ gj}


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Theorem 1:

Let G be a finite group of order |G|i.e., G = {g1, g2, . . . , gi, . . . , gj , . . . , g|G|}.

Let m and n be any two positive integers such that(m, |G|) = (n, |G|) = 1.

Define |G| × |G| matrix A as A = [gmi .gnj ]; i, j ∈ {1, 2, 3, , |G|}.Then A is a Latin square of order |G|.


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Theorem 2:

Let a and b be any two fixed elements of a finite group G.

Define A = [(a ∗ gi) ∗ (gj ∗ b)], B = [(a ∗ gi) ∗ (b ∗ gj)],C = [(gi ∗ a) ∗ (gj ∗ b)] and D = [(gi ∗ a) ∗ (b ∗ gj)].

Then A,B,C and D are all Latin squares of order |G|.


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Theorem 3:

Let T1, T2 be any two automorphisms of a finite group of G.

Define |G| × |G| matrix A as A = [T1(gi) ∗ T2(gj)].

Then A is a Latin square of order |G|.


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Let n be any positive integer

Zn = {0, 1, . . . , n− 1}, the set of integers modulo n

+n and ×n is the addition and multiplication (mod n) defined onZn

Define Z∗n = Zn − {0} with a lies in Z

∗n if and only if (a, n) = 1


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select a, b lies in Z∗n with [(a, n) = (b, n) = 1]

select r and s lies in Zn

Define a bijective map fa,r : Zn → Zn

by fa,r(x) = a.x+ r( mod n)

similarly, define a bijective map gb,s : Zn → Zn

by gb,s(x) = b.x+ s( mod n)


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Define A = [fa,r(i) + ga,r(j)( mod n)]

A is a Latin square of order n based on Zn

By simple corollary, A = [(i+ j)( mod n)] a Latin square of order nbased on Zn

B = [(a.i+ j)( mod n)] a Latin square of order n based on Zn


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Let m = 2n+ 1

Construct an m×m matrix C as followsC = [cij ] = [(n+ 1)(i+ j)( mod m)]

C is a Latin square of order m based on Zm

This is the beautiful example of idempotent symmetric Latinsquare of odd order


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For any fixed non negative integer k, m = 2k+1 + 5

Construct m×m matrix D as followsD = [dij ] = [(2k+3)(i+ j)( mod m)]for i, j ∈ {0, 1, 2, . . . , 2k+1 + 4}

This D also an idempotent symmetric Latin square of order mbased on Zm


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Let us construct more generally idempotent symmectric Latinsquare

Let m = 2k+1 + p, where p is odd prime

Define a matrix of order m byX = [xij ] = [[2k + (p+ 1/2)](i+ j)( mod m)]i, j ∈ {0, 1, 2, . . . , 2k+1 + p− 1}

X is idempotent symmetric Latin square of order m based on Zm


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Let p be any given odd prime

Let us select a positive integer e such that (e, p− 1) = 1

Define a matrix X of order p× p as followsX = [xij ] = [(p+ 1/2).(ie + je)( mod p)]i, j ∈ {0, 1, 2, . . . , p− 1} is a symmetric Latin square


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Construct an RSA public key cryptography Latin square

Select two distinct very large odd primes p, q such that n = pq

Select two fixed positive integers e, d such that(e, (p− 1)(q − 1)) = 1 and (d, (p− 1)(q − 1)) = 1

Now define a matrix A = [ie + jd( mod n)]

A is a Latin square of order n based on Zn

B = [(aie + j)( mod n)] where (a, pq) = 1

Then B is a latin square of order n ( Here a, e are fixed )


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REFERENCES

Adi Shamir, (1979), How to share a secret, Communications of theACM 22 (11) 612-613.

Asmuth, C., Bloom, J.: A modular approach to key safeguarding.IEEE Trans. inform. Theory, 29 (1983) 208Öš10.

S. Barnard, J.M. Child, Higher Algebra, The Macmillan and Co.,1952.

R. Balakrishnan and K. Ranganathan, A textbook of Graph Theory,Springer, Berlin, 2000.


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REFERENCES

Beimel. A, Secret-sharing schemes: a survey, Proceedings of theThird international conference on Coding and cryptology, Berlin,Heidelberg, 2011, Springer-Verlag, IWCC’11, pages 11-46.

E.R.Berlekamp, Algebraic Coding Theory, NY, McGraw-Hill, 1968.

Blakley, G. R. (1979), Safeguarding cryptographic keys,Proceedings of the National Computer Conference 48, 313-317.


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REFERENCES

I. N. Herstein, Topics in Algebra, 2nd Edition, Wiley, 1975.

Chandramowliswaran. N,Srinivasan. S,Muralikrishna. P,Authenticated key distribution using given set of primes for secretsharing, Journal of Systems Science and Control Engineering(Taylor and Francis), No 3 (2015) 106-112.

Srinivasan. S, Muralikrishna. P, Chandramowliswaran. N,Authenticated Multiple Key Distribution using Simple ContinuedFraction, International Journal of Pure and Applied Mathematics,87 No 2 (2013) 349-354.


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REFERENCES

Muralikrishna. P, Srinivasan. S , Chandramowliswaran. N, SecureSchemes for Secret Sharing and Key Distribution using Pell’sequation, International Journal of Pure and Applied Mathematics,85 No 5 (2013) 933-937.

Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, AnIntroduction to the Theory of Numbers, John Wiley.

Tom M. Apostol, Introduction to Analytic Number Theory, Springer.


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