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Cryptography via Burnside Groups Antonio R. Nicolosi Stevens Institute of Technology Based on work w/ G.Baumslag, N.Fazio, K.Iga, L.Perret, V.Shpilrain and W.E.Skeith III Mathematics of Cryptography September 1, 2015. University of California,

  1. Cryptography via Burnside Groups Antonio R. Nicolosi Stevens Institute of Technology Based on work w/ G.Baumslag, N.Fazio, K.Iga, L.Perret, V.Shpilrain and W.E.Skeith III Mathematics of Cryptography September 1, 2015. University of California, Irvine, CA

  2. Talk Preview Goal Identify viable intractability assumptions from combinatorial group theory Evidence of (average-case) hardness (random self-reducibility) Cryptographically useful Approach Generalize well-established crypto assumptions (LPN/LWE) to a group-theoretic setting Study instantiation in suitable non-commutative groups Antonio R. Nicolosi Cryptography via Burnside Groups

  3. Outline Background 1 Burnside Groups ( B n ) Learning Burnside Homomorphisms with Noise ( B n -LHN) Random Self-Reducibility of B n -LHN 2 Cryptography (Minicrypt) via Burnside Groups 3 Antonio R. Nicolosi Cryptography via Burnside Groups

  4. Outline Background 1 Burnside Groups ( B n ) Learning Burnside Homomorphisms with Noise ( B n -LHN) Random Self-Reducibility of B n -LHN 2 Cryptography (Minicrypt) via Burnside Groups 3 Antonio R. Nicolosi Cryptography via Burnside Groups

  5. Burnside Problem (Informal) Are groups whose elements all have finite order necessarily finite ? What is their combinatorial structure? Antonio R. Nicolosi Cryptography via Burnside Groups

  6. Free Burnside group of exponent m B ( n , m ) : “Most generic” group with n generators where the order of all elements divides m Generators x 1 , . . . , x n (like indeterminates in a multivariate poly) Elements are sequences of x i and x − 1 i Empty sequence is the identity element of the group Exponent condition: For every w ∈ B ( n , m ) it holds that w m = 1 Examples: x 1 x − 1 x − 1 1 x − 1 4 x 1 ∈ B ( 4 , 3 ) , ∈ B ( 4 , 3 ) 4 x 2 1 = x − 1 1 , but x 1 x − 1 4 x 1 � = x − 1 1 x − 1 = x 1 x 1 x − 1 ( B ( 4 , 3 ) is not abelian) 4 4 On the other hand: 4 ) 3 = 1 x 1 x − 1 4 x 1 = x 4 x − 1 since x 1 x − 1 4 x 1 x − 1 4 x 1 x − 1 = ( x 1 x − 1 1 x 4 , 4 Antonio R. Nicolosi Cryptography via Burnside Groups

  7. Free Burnside group of exponent m B ( n , m ) : “Most generic” group with n generators where the order of all elements divides m Generators x 1 , . . . , x n (like indeterminates in a multivariate poly) Elements are sequences of x i and x − 1 i Empty sequence is the identity element of the group Exponent condition: For every w ∈ B ( n , m ) it holds that w m = 1 Examples: x 1 x − 1 x − 1 1 x − 1 4 x 1 ∈ B ( 4 , 3 ) , ∈ B ( 4 , 3 ) 4 x 2 1 = x − 1 1 , but x 1 x − 1 4 x 1 � = x − 1 1 x − 1 = x 1 x 1 x − 1 ( B ( 4 , 3 ) is not abelian) 4 4 On the other hand: 4 ) 3 = 1 x 1 x − 1 4 x 1 = x 4 x − 1 since x 1 x − 1 4 x 1 x − 1 4 x 1 x − 1 = ( x 1 x − 1 1 x 4 , 4 Antonio R. Nicolosi Cryptography via Burnside Groups

  8. Free Burnside group of exponent m B ( n , m ) : “Most generic” group with n generators where the order of all elements divides m Generators x 1 , . . . , x n (like indeterminates in a multivariate poly) Elements are sequences of x i and x − 1 i Empty sequence is the identity element of the group Exponent condition: For every w ∈ B ( n , m ) it holds that w m = 1 Examples: x 1 x − 1 x − 1 1 x − 1 4 x 1 ∈ B ( 4 , 3 ) , ∈ B ( 4 , 3 ) 4 x 2 1 = x − 1 1 , but x 1 x − 1 4 x 1 � = x − 1 1 x − 1 = x 1 x 1 x − 1 ( B ( 4 , 3 ) is not abelian) 4 4 On the other hand: 4 ) 3 = 1 x 1 x − 1 4 x 1 = x 4 x − 1 since x 1 x − 1 4 x 1 x − 1 4 x 1 x − 1 = ( x 1 x − 1 1 x 4 , 4 Antonio R. Nicolosi Cryptography via Burnside Groups

  9. Free Burnside group of exponent m B ( n , m ) : “Most generic” group with n generators where the order of all elements divides m Generators x 1 , . . . , x n (like indeterminates in a multivariate poly) Elements are sequences of x i and x − 1 i Empty sequence is the identity element of the group Exponent condition: For every w ∈ B ( n , m ) it holds that w m = 1 Examples: x 1 x − 1 x − 1 1 x − 1 4 x 1 ∈ B ( 4 , 3 ) , ∈ B ( 4 , 3 ) 4 x 2 1 = x − 1 1 , but x 1 x − 1 4 x 1 � = x − 1 1 x − 1 = x 1 x 1 x − 1 ( B ( 4 , 3 ) is not abelian) 4 4 On the other hand: 4 ) 3 = 1 x 1 x − 1 4 x 1 = x 4 x − 1 since x 1 x − 1 4 x 1 x − 1 4 x 1 x − 1 = ( x 1 x − 1 1 x 4 , 4 Antonio R. Nicolosi Cryptography via Burnside Groups

  10. Free Burnside group of exponent m B ( n , m ) : “Most generic” group with n generators where the order of all elements divides m Generators x 1 , . . . , x n (like indeterminates in a multivariate poly) Elements are sequences of x i and x − 1 i Empty sequence is the identity element of the group Exponent condition: For every w ∈ B ( n , m ) it holds that w m = 1 Examples: x 1 x − 1 x − 1 1 x − 1 4 x 1 ∈ B ( 4 , 3 ) , ∈ B ( 4 , 3 ) 4 x 2 1 = x − 1 1 , but x 1 x − 1 4 x 1 � = x − 1 1 x − 1 = x 1 x 1 x − 1 ( B ( 4 , 3 ) is not abelian) 4 4 On the other hand: 4 ) 3 = 1 x 1 x − 1 4 x 1 = x 4 x − 1 since x 1 x − 1 4 x 1 x − 1 4 x 1 x − 1 = ( x 1 x − 1 1 x 4 , 4 Antonio R. Nicolosi Cryptography via Burnside Groups

  11. Free Burnside group of exponent m B ( n , m ) : “Most generic” group with n generators where the order of all elements divides m Generators x 1 , . . . , x n (like indeterminates in a multivariate poly) Elements are sequences of x i and x − 1 i Empty sequence is the identity element of the group Exponent condition: For every w ∈ B ( n , m ) it holds that w m = 1 Examples: x 1 x − 1 x − 1 1 x − 1 4 x 1 ∈ B ( 4 , 3 ) , ∈ B ( 4 , 3 ) 4 x 2 1 = x − 1 1 , but x 1 x − 1 4 x 1 � = x − 1 1 x − 1 = x 1 x 1 x − 1 ( B ( 4 , 3 ) is not abelian) 4 4 On the other hand: 4 ) 3 = 1 x 1 x − 1 4 x 1 = x 4 x − 1 since x 1 x − 1 4 x 1 x − 1 4 x 1 x − 1 = ( x 1 x − 1 1 x 4 , 4 Antonio R. Nicolosi Cryptography via Burnside Groups

  12. Free Burnside group of exponent m B ( n , m ) : “Most generic” group with n generators where the order of all elements divides m Generators x 1 , . . . , x n (like indeterminates in a multivariate poly) Elements are sequences of x i and x − 1 i Empty sequence is the identity element of the group Exponent condition: For every w ∈ B ( n , m ) it holds that w m = 1 Examples: x 1 x − 1 x − 1 1 x − 1 4 x 1 ∈ B ( 4 , 3 ) , ∈ B ( 4 , 3 ) 4 x 2 1 = x − 1 1 , but x 1 x − 1 4 x 1 � = x − 1 1 x − 1 = x 1 x 1 x − 1 ( B ( 4 , 3 ) is not abelian) 4 4 On the other hand: 4 ) 3 = 1 x 1 x − 1 4 x 1 = x 4 x − 1 since x 1 x − 1 4 x 1 x − 1 4 x 1 x − 1 = ( x 1 x − 1 1 x 4 , 4 Antonio R. Nicolosi Cryptography via Burnside Groups

  13. Free Burnside group of exponent m B ( n , m ) : “Most generic” group with n generators where the order of all elements divides m Generators x 1 , . . . , x n (like indeterminates in a multivariate poly) Elements are sequences of x i and x − 1 i Empty sequence is the identity element of the group Exponent condition: For every w ∈ B ( n , m ) it holds that w m = 1 Examples: x 1 x − 1 x − 1 1 x − 1 4 x 1 ∈ B ( 4 , 3 ) , ∈ B ( 4 , 3 ) 4 x 2 1 = x − 1 1 , but x 1 x − 1 4 x 1 � = x − 1 1 x − 1 = x 1 x 1 x − 1 ( B ( 4 , 3 ) is not abelian) 4 4 On the other hand: 4 ) 3 = 1 x 1 x − 1 4 x 1 = x 4 x − 1 since x 1 x − 1 4 x 1 x − 1 4 x 1 x − 1 = ( x 1 x − 1 1 x 4 , 4 Antonio R. Nicolosi Cryptography via Burnside Groups

  14. Burnside Groups (cont’d) Characterizing B ( n , m ) not so easy . . . Finite and abelian, isomorphic to ( F n B ( n , 2 ) 2 , +) Finite, non-commutative, much larger than ( F n B ( n , 3 ) 3 , +) B ( n , 4 ) Finite B ( n , 5 ) Unknown B ( n , 6 ) Finite B ( n , 7 ) Unknown . . . . . . B ( n , m ) , m “large” Infinite Will focus on B ( n , 3 ) (simplest case beyond vector spaces) Notation: B n . = B ( n , 3 ) Antonio R. Nicolosi Cryptography via Burnside Groups

  15. Burnside Groups (cont’d) Characterizing B ( n , m ) not so easy . . . Finite and abelian, isomorphic to ( F n B ( n , 2 ) 2 , +) Finite, non-commutative, much larger than ( F n B ( n , 3 ) 3 , +) B ( n , 4 ) Finite B ( n , 5 ) Unknown B ( n , 6 ) Finite B ( n , 7 ) Unknown . . . . . . B ( n , m ) , m “large” Infinite Will focus on B ( n , 3 ) (simplest case beyond vector spaces) Notation: B n . = B ( n , 3 ) Antonio R. Nicolosi Cryptography via Burnside Groups

  16. B n : Burnside Groups of Exponent 3 B n : “Most generic” group with n generators where the order of all non-identity elements is 3 Generators x 1 , . . . , x n Elements are sequences of x i and x − 1 i Exponent condition: ∀ w ∈ B n , www = 1 ( ⋆ ) Q : “Most generic”!? A : The only non-trivial identities in B n are those implied by ( ⋆ ) ⇒ B n non-commutative x i x j � = x j x i for any two distinct generators ( i � = j ) ⇒ Group operation in B n defined “formally” To “multiply” w 1 , w 2 ∈ B n , just concatenate them Simplifications may arise at the interface of w 1 and w 2 Antonio R. Nicolosi Cryptography via Burnside Groups

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