1. bibliography
Combinatorial conditions for secret sharing
for Public Key Cryptography
by
Dr. N. Chandramowliswaran
Professor
Applied Sciences, NCU, Gurgaon
FEB. 27, 2016
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 1 / 19

2. bibliography
Latin Squares
Definition 1
Let n be a given positive integer and S = {s1, s2, . . . , sn} be a given set
of n distinct elements.
A Latin square of order n based on S is an n × n array in which every
element of S occurs exactly once in each row and exactly once in each
column.
Thus each of the rows and columns of a Latin square is a permutation
of S.
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 2 / 19

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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 varies
from 1 to n
such a function f is called a Latin function defined on G × G to G
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 3 / 19

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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 of
A = {gi ∗ g1, gi ∗ g2, . . . , gi ∗ gi, . . . , gi ∗ gj, . . . , gi ∗ gn} and
the j-th column of
A = {g1 ∗ gj, g2 ∗ gj, . . . , gi ∗ gj, . . . , gj ∗ gj, . . . , gn ∗ gj}
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 4 / 19

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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 = [gm
i .gn
j ]; i, j ∈ {1, 2, 3, , |G|}.
Then A is a Latin square of order |G|.
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 5 / 19

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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|.
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 6 / 19

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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|.
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 7 / 19

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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 on
Zn
Define Z∗
n = Zn − {0} with a lies in Z∗
n if and only if (a, n) = 1
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 8 / 19

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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)
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 9 / 19

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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 n
based on Zn
B = [(a.i + j)( mod n)] a Latin square of order n based on Zn
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 10 / 19

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Let m = 2n + 1
Construct an m × m matrix C as follows
C = [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 Latin
square of odd order
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 11 / 19

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For any fixed non negative integer k, m = 2k+1 + 5
Construct m × m matrix D as follows
D = [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 m
based on Zm
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 12 / 19

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Let us construct more generally idempotent symmectric Latin
square
Let m = 2k+1 + p, where p is odd prime
Define a matrix of order m by
X = [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
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 13 / 19

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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 follows
X = [xij] = [(p + 1/2).(ie + je)( mod p)]
i, j ∈ {0, 1, 2, . . . , p − 1} is a symmetric Latin square
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 14 / 19

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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 )
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 15 / 19

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REFERENCES
Adi Shamir, (1979), How to share a secret, Communications of the
ACM 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.
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 16 / 19

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REFERENCES
Beimel. A, Secret-sharing schemes: a survey, Proceedings of the
Third 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.
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 17 / 19

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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 secret
sharing, 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 Continued
Fraction, International Journal of Pure and Applied Mathematics,
87 No 2 (2013) 349-354.
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 18 / 19

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REFERENCES
Muralikrishna. P, Srinivasan. S , Chandramowliswaran. N, Secure
Schemes for Secret Sharing and Key Distribution using Pell’s
equation, International Journal of Pure and Applied Mathematics,
85 No 5 (2013) 933-937.
Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An
Introduction to the Theory of Numbers, John Wiley.
Tom M. Apostol, Introduction to Analytic Number Theory, Springer.
Dr. N. Chandramowliswaran (NCU) INVITED TALK FEB. 27, 2016 19 / 19