Motivating Scenario I Card-based Cryptographic Protocols Using a Minimal Number of Cards Alexander Koch, Stefan Walzer, Kevin Härtel [asiacrypt/KochWH15] DEPARTMENT OF INFORMATICS, INSTITUTE OF THEORETICAL INFORMATICS 1 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards KIT – University of the State of Baden-Wuerttemberg and www.kit.edu National Research Center of the Helmholtz Association
Motivating Scenario I Secrets: Do I love him/her? To compute: Is there mutual affection? � Secure 2-party AND without computers Trusted Computation 1 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
Motivating Scenario I Secrets: Do I love him/her? To compute: Is there mutual affection? � Secure 2-party AND without computers Trusted Computation 1 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
Motivating Scenario I Secrets: Do I love him/her? To compute: Is there mutual affection? � Secure 2-party AND without computers Trusted Computation 1 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
Motivating Scenario I Secrets: Do I love him/her? To compute: Is there mutual affection? � Secure 2-party AND without computers Trusted Computation 1 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
Motivating Scenario II Hey, help me compute y d mod n . 2 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
Motivating Scenario II Hey, help me compute y d mod n . Sure, just tell me... 2 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
Motivating Scenario II Hey, help me compute y d mod n . Sure, just tell me... I’m not telling you y , d or n . 2 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
Motivating Scenario II Hey, help me compute y d mod n . Sure, just tell me... I’m not telling you y , d or n . Nor may you know the result. 2 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
Motivating Scenario II Hey, help me compute y d mod n . Sure, just tell me... I’m not telling you y , d or n . Nor may you know the result. ... 2 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
Motivating Scenario II Hey, help me compute y d mod n . Sure, just tell me... I’m not telling you y , d or n . Nor may you know the result. Sure, I’ll get some cards. 2 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
Setting and Goal Two types of indistinguishable cards: Heart ♥ and club ♣ with backside . Encode bits as ♣ ♥ ˆ = 0 = 1 ♥ ♣ ˆ Our goal (“committed format”) Take face-down input (bits a , b ) ( a ∧ b ) Compute face-down output Learn nothing about the input or output during protocol run. 3 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
Setting and Goal Two types of indistinguishable cards: Heart ♥ and club ♣ with backside . Encode bits as ♣ ♥ ˆ = 0 = 1 ♥ ♣ ˆ Our goal (“committed format”) Take face-down input (bits a , b ) ( a ∧ b ) Compute face-down output Learn nothing about the input or output during protocol run. 3 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
The if-then-else Operator Definition � b if a = 1 (if a then b else c) : = c if a = 0 Also known as: (a ? b : c) Note: (a ∧ b) ≡ (if a then b else 0) (if a then b else c) ≡ (if ¬ a then c else b) 4 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
Computing “if a then b else c” (cmp. [faw/MizukiS09]) Conceptually With Cards Input: a,b,c Input: � �� � � �� � � �� � a c b With equal probability do With equal probability set ( a ′ , b ′ , c ′ ) = ( a , b , c ) or either nothing or ( a ′ , b ′ , c ′ ) = ( ¬ a , c , b ) Test a ′ Turn 1,2 1 0 ♥ ♣ ♣ ♥ return b’ return c’ ♥ ♣ ♣ ♥ � �� � � �� � output output 5 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
Can we do better than six cards? Main Question: How many cards needed to compute a ∧ b where Input and output encoded as ♥ ♣ = 1, ♣ ♥ = 0. We are and remain oblivious of input and output. Our Results 4 cards 5 cards Not yet published probably 6 cards 4 cards 6 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
Can we do better than six cards? Main Question: How many cards needed to compute a ∧ b where Input and output encoded as ♥ ♣ = 1, ♣ ♥ = 0. We are and remain oblivious of input and output. Our Results 4 cards 5 cards Not yet published probably 6 cards 4 cards 6 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
Can we do better than six cards? Main Question: How many cards needed to compute a ∧ b where Input and output encoded as ♥ ♣ = 1, ♣ ♥ = 0. We are and remain oblivious of input and output. Our Results 4 cards (Model of M izuki & S hizuya) 5 cards ( MS but a-priori bound runtime) Not yet published probably 6 cards ( MS but only “uniform closed” shuffles) 4 cards (Player-Perm model) 6 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
Computational Model Based on ijisec/MizukiS14 Operations (perm , π ). Apply permutation π to the sequence of cards. (shuffle , Π , F ). Apply permutation π ∈ Π , drawn according to F . Note : We don’t know which π was chosen! (turn , T ). Reveal cards in positions given by T . (result , b 1 , b 2 ). Output cards in positions b 1 , b 2 . Correctness: Cards given by result-operation always encodes correct output bit. Security: The observations (made during turns) are stochastially independent of input and output. 7 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
State Transitions: The Six-Card Protocol Protocol State: ♥♣♥♣♣♥ X 11 Annotate currently possible ♥♣♣♥♣♥ X 10 sequences with probability in ♣♥♥♣♣♥ X 01 terms of symbolic input prob. ♣♥♣♥♣♥ X 00 X ij = Pr [ a = i , b = j ] 8 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
State Transitions: The Six-Card Protocol Protocol State: ♥♣♥♣♣♥ X 11 Annotate currently possible ♥♣♣♥♣♥ X 10 sequences with probability in ♣♥♥♣♣♥ X 01 terms of symbolic input prob. ♣♥♣♥♣♥ X 00 X ij = Pr [ a = i , b = j ] 8 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
State Transitions: The Six-Card Protocol Protocol State: ♥♣♥♣♣♥ X 11 Annotate currently possible ♥♣♣♥♣♥ X 10 sequences with probability in ♣♥♥♣♣♥ X 01 terms of symbolic input prob. ♣♥♣♥♣♥ X 00 X ij = Pr [ a = i , b = j ] 8 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
State Transitions: The Six-Card Protocol Protocol State: ♥♣♥♣♣♥ X 11 Annotate currently possible ♥♣♣♥♣♥ X 10 sequences with probability in ♣♥♥♣♣♥ X 01 terms of symbolic input prob. ♣♥♣♥♣♥ X 00 X ij = Pr [ a = i , b = j ] ( shuffle , { id , ( 1 2 )( 3 5 )( 4 6 ) } ) ♥♣♥♣♣♥ 1 / 2 X 11 ♥♣♣♥♣♥ 1 2 X 10 + 1 / / 2 X 00 ♣♥♥♣♣♥ 1 / 2 X 01 ♣♥♣♥♣♥ 1 2 X 00 + 1 / / 2 X 10 ♣♥♣♥♥♣ 1 / 2 X 11 ♥♣♣♥♥♣ 1 / 2 X 01 8 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
State Transitions: The Six-Card Protocol Protocol State: ♥♣♥♣♣♥ X 11 Annotate currently possible ♥♣♣♥♣♥ X 10 sequences with probability in ♣♥♥♣♣♥ X 01 terms of symbolic input prob. ♣♥♣♥♣♥ X 00 X ij = Pr [ a = i , b = j ] ( shuffle , { id , ( 1 2 )( 3 5 )( 4 6 ) } ) ♥♣♥♣♣♥ 1 / 2 X 11 ♥♣♣♥♣♥ 1 2 X 10 + 1 / / 2 X 00 ♣♥♥♣♣♥ 1 / 2 X 01 ♣♥♣♥♣♥ 1 2 X 00 + 1 / / 2 X 10 ♣♥♣♥♥♣ 1 / 2 X 11 ♥♣♣♥♥♣ 1 / 2 X 01 8 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
State Transitions: The Six-Card Protocol Protocol State: ♥♣♥♣♣♥ X 11 Annotate currently possible ♥♣♣♥♣♥ X 10 sequences with probability in ♣♥♥♣♣♥ X 01 terms of symbolic input prob. ♣♥♣♥♣♥ X 00 X ij = Pr [ a = i , b = j ] ( shuffle , { id , ( 1 2 )( 3 5 )( 4 6 ) } ) ♥♣♥♣♣♥ 1 / 2 X 11 ♥♣♣♥♣♥ 1 2 X 10 + 1 / / 2 X 00 ♣♥♥♣♣♥ 1 / 2 X 01 ♣♥♣♥♣♥ 1 2 X 00 + 1 / / 2 X 10 ♣♥♣♥♥♣ 1 / 2 X 11 ♥♣♣♥♥♣ 1 / 2 X 01 8 2015-12-03 Koch, Walzer - Card-based Cryptographic Protocols Using a Minimal Number of Cards
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