Key Exchange and Encryption SchemesBased on commutative rings
N.ChandramowliswaranProfessor
Indian Institute of ManagementIndore - India.
Jan 23, 2014
Indore IIM
Cryptography 1
Secret Key Sharing using Finite Groups
Step 1 Let P = 2pr + 1 and Q = 2qs + 1,where P,Q, p and q are very large odd primes(which is kept secret)
Indore IIM
Cryptography 2
Secret Key Sharing using Finite Groups
Step 3 Define G = {1 ≤ x ≤ N ∣ (x,N) = 1}
Indore IIM
Cryptography 2
Secret Key Sharing using Finite Groups
Step 4 Let ×N be the multiplication modulo N.Clearely (G,×N) forms a finite group withO(G) = �(N) = 4prqs
Indore IIM
Cryptography 2
Secret Key Sharing using Finite Groups
Step 5 Let s (given secret) be the element of G
Indore IIM
Cryptography 2
Secret Key Sharing using Finite Groups
Step 6 From finite group theory, any mapΨ, g 7−→ gm is always an automorphism of G,if (m,O(G)) = 1
Indore IIM
Cryptography 2
Secret Key Sharing using Finite Groups
Step 7 Let m = ℓ1 + ℓ2 + ⋅ ⋅ ⋅+ ℓt.Consider s = xm
s = xℓ1+ℓ2+⋅⋅⋅+ℓt
s = xℓ1xℓ2 . . . xℓt
s = y1y2 . . . yt
where yi = xℓi(mod N), 1 ≤ i ≤ t be the individualshare holders.
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Cryptography 2
For any x, y ∈ R, x = (x1, x2, . . . , xN−1, 0), y = (y1, y2, . . . , yN−1, 0)then definex ⊕ y =((x1 + y1) mod N, (x2 + y2) mod N − 1, . . . , (xN−1 + yN−1) mod 2, 0)x ⊗ y = ((x1y1) mod N, (x2y2) mod N − 1, . . . , (xN−1yN−1) mod 2, 0)
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Cryptography 3
Theorem 1 R forms a commutative ring with unity with respectto the addition and multiplication defined above, having N!elements.
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Cryptography 4
In this Ring R the unique unity element is (1, 1, 1, . . . , 1, 0). Here1 appears exactly N − 1 times.
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Cryptography 4
Let U(R) = {(a(1,N), a(2,N−1), a(3,N−2), . . . , a(N−1,2) = 1, 0)},Where G.C.D (a(i,N−i+1),N − i + 1) = 1 for alli, i = {1, 2, 3, . . . ,N − 1}. Also 1 ≤ a(i,N−i+1) ≤ N − i, then(R) ∣= �(N)�(N − 1)�(N − 2) . . . �(2).
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Cryptography 4
Definition: Indicator of an element with respect to a subgroupH of a finite group GIf H is a subgroup of a finite group G, then for any element a inG there is an integer n such that an ∈ H. If a is already in H wesimply take n = 1. If a /∈ H we can take n to be the order of a,since an = e ∈ H. However, there may be a smaller positivepower of a which lies in H. By the well ordering principle thereis a smallest positive integer n such that an ∈ H. We can callthis integer the indicator of a in H.
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Cryptography 5
Theorem 2 Let H be a subgroup of a finite abelian group G,where H ∕= G. Choose an element a in G, a /∈ H, and let h bethe indicator of a in H. Then the set of products,K = {xak : x ∈ H and k = 0, 1, 2, . . . , h − 1} is a subgroup of Gwhich contains H. Moreover, the order of K is h times that of H,i.e., ∣ K ∣= h ∣ H ∣ .
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Cryptography 6
Application of this Theorem in Cryptography
Let G be a given finite abelian group such that ∣ G ∣ is known tothe public
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Cryptography 7
Application of this Theorem in Cryptography
Let H be the non-trivial proper (secret) subgroup of G with ∣ H ∣kept secret
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Cryptography 7
Application of this Theorem in Cryptography
Represent the given message m ∕= 1 ∈ G with m /∈ H
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Cryptography 7
Application of this Theorem in Cryptography
Compute the Indicator of m with respect to H = h, kept secret.
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Cryptography 7
Application of this Theorem in Cryptography
Define a subgroup K of G (K is also kept secret)K = {xmk : x ∈ H and k = 0, 1, 2, . . . , h − 1}Then clearly K is a subgroup of G which contains H.∣ K ∣= h ∣ H ∣, ∣ K ∣ is kept secret.m ∈ K
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Cryptography 7
Application of this Theorem in Cryptography
Select an integer � such that G.C.D (� , h ∣ H ∣) = 1
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Cryptography 7
Application of this Theorem in Cryptography
Now the encryption of the message m is m� ∈ K
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Cryptography 7
Application of this Theorem in Cryptography
Since (�, h ∣ H ∣) = 1, there is a unique � such that�� ≡ 1 (mod h ∣ H ∣)
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Cryptography 7
Application of this Theorem in Cryptography
The decrypted message is m�� ≡ m (mod K)
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Cryptography 7
LetH = {(a(1,N), a(2,N−1), a(3,N−2), . . . , a(r,N−r+1) = 1, . . . , a(s,N−s+1) =1,. . . , a(N−1,2) = 1, 0)},
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Cryptography 9
Let m = ℓ⊕ v (m /∈ H)m = ℓ⊕ v = (p1, p2, . . . , pN−1, 0)⊕ (q1, q2, . . . , qN−1, 0)
i.e., ℓ⊕ v = (p1 + q1, p2 + q2, . . . , pN−1 + qN−1, 0) ∈ K
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Cryptography 9
Proposition 2.1 Let G be any given finite group and let G.C.D(r, ∣ G ∣) = 1. Then the map sending g to gr is always apermutation on G.
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Cryptography 10
Proposition 2.2 If N is a Normal subgroup of finite index in agroup G, and H is subgroup of finite order in G with G.C.D(∣ (G : N) ∣, ∣ H ∣) = 1, then H lies in N ( H is a Subgroup of N).
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Cryptography 10
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 keysafeguarding. IEEE Trans. inform. Theory, 29 (1983)208U210.
S. Barnard, J.M. Child, Higher Algebra, The Macmillan andCo., 1952.
R. Balakrishnan and K. Ranganathan, A textbook of GraphTheory, Springer, Berlin, 2000.
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Cryptography 12
REFERENCES
Beimel. A, Secret-sharing schemes: a survey, Proceedingsof the Third international conference on Coding andcryptology, 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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Cryptography 13
REFERENCES
I. N. Herstein, Topics in Algebra, 2nd Edition, Wiley, 1975.
Mignotte, M.: How to share a secret. Advances inCryptology U EurocryptŠ82, LNCS, Springer-Verlag, 149(1983) 371-375.
Muralikrishna. P, Srinivasan. S , Chandramowliswaran. N,Secure Schemes for Secret Sharing and Key Distributionusing Pell’s equation, International Journal of Pure andApplied Mathematics, 85 No 5 (2013) 933-937.
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Cryptography 14
REFERENCES
Srinivasan. S, Muralikrishna. P, Chandramowliswaran. N,Authenticated Multiple Key Distribution using SimpleContinued Fraction, International Journal of Pure andApplied Mathematics, 87 No 2 (2013) 349-354.
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.
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Cryptography 15