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The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta University of Salerno joint work (in progress) with Antonio Tortora Groups St Andrews 2017 in Birmingham The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017

Background Background In cryptography, one of the most studied problems is how to share a secret key over an insecure channel. Key exchange methods are usually based on one-way functions, that is functions which are easy to compute but whose inverses are difficult to determine. The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017

Background Background In cryptography, one of the most studied problems is how to share a secret key over an insecure channel. Key exchange methods are usually based on one-way functions, that is functions which are easy to compute but whose inverses are difficult to determine. The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017

Background There are several ways in which group theory can be used to construct one-way functions. In 1999, I. Anshel, M. Anshel and D. Goldfeld introduced a key ex- change protocol whose platform is a nonabelian group G . The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017

Background There are several ways in which group theory can be used to construct one-way functions. In 1999, I. Anshel, M. Anshel and D. Goldfeld introduced a key ex- change protocol whose platform is a nonabelian group G . The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017

Anshel-Anshel-Goldfeld Anshel-Anshel-Goldfeld Circumstances: Alice and Bob want to agree on a common key. Platform: let G be a nonabelian group PUBLIC KEYS Alice chooses a 1 , . . . , a l in G and makes them PUBLIC. Bob chooses b 1 , . . . , b k in G and makes them PUBLIC. PRIVATE KEYS Alice chooses A ∈ � a 1 , . . . , a l � . Bob chooses B ∈ � b 1 , . . . , b k � . EXCHANGED INFORMATION Alice computes b ′ 1 , . . . , b ′ 1 = b A k = b A k , and sends them to Bob. Bob computes a ′ 1 , . . . , a ′ 1 = a B l = a B l , and sends them to Alice. The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017

Anshel-Anshel-Goldfeld Anshel-Anshel-Goldfeld Circumstances: Alice and Bob want to agree on a common key. Platform: let G be a nonabelian group PUBLIC KEYS Alice chooses a 1 , . . . , a l in G and makes them PUBLIC. Bob chooses b 1 , . . . , b k in G and makes them PUBLIC. PRIVATE KEYS Alice chooses A ∈ � a 1 , . . . , a l � . Bob chooses B ∈ � b 1 , . . . , b k � . EXCHANGED INFORMATION Alice computes b ′ 1 , . . . , b ′ 1 = b A k = b A k , and sends them to Bob. Bob computes a ′ 1 , . . . , a ′ 1 = a B l = a B l , and sends them to Alice. The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017

Anshel-Anshel-Goldfeld Anshel-Anshel-Goldfeld Circumstances: Alice and Bob want to agree on a common key. Platform: let G be a nonabelian group PUBLIC KEYS Alice chooses a 1 , . . . , a l in G and makes them PUBLIC. Bob chooses b 1 , . . . , b k in G and makes them PUBLIC. PRIVATE KEYS Alice chooses A ∈ � a 1 , . . . , a l � . Bob chooses B ∈ � b 1 , . . . , b k � . EXCHANGED INFORMATION Alice computes b ′ 1 , . . . , b ′ 1 = b A k = b A k , and sends them to Bob. Bob computes a ′ 1 , . . . , a ′ 1 = a B l = a B l , and sends them to Alice. The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017

Anshel-Anshel-Goldfeld Anshel-Anshel-Goldfeld Circumstances: Alice and Bob want to agree on a common key. Platform: let G be a nonabelian group PUBLIC KEYS Alice chooses a 1 , . . . , a l in G and makes them PUBLIC. Bob chooses b 1 , . . . , b k in G and makes them PUBLIC. PRIVATE KEYS Alice chooses A ∈ � a 1 , . . . , a l � . Bob chooses B ∈ � b 1 , . . . , b k � . EXCHANGED INFORMATION Alice computes b ′ 1 , . . . , b ′ 1 = b A k = b A k , and sends them to Bob. Bob computes a ′ 1 , . . . , a ′ 1 = a B l = a B l , and sends them to Alice. The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017

Anshel-Anshel-Goldfeld Anshel-Anshel-Goldfeld Circumstances: Alice and Bob want to agree on a common key. Platform: let G be a nonabelian group PUBLIC KEYS Alice chooses a 1 , . . . , a l in G and makes them PUBLIC. Bob chooses b 1 , . . . , b k in G and makes them PUBLIC. PRIVATE KEYS Alice chooses A ∈ � a 1 , . . . , a l � . Bob chooses B ∈ � b 1 , . . . , b k � . EXCHANGED INFORMATION Alice computes b ′ 1 , . . . , b ′ 1 = b A k = b A k , and sends them to Bob. Bob computes a ′ 1 , . . . , a ′ 1 = a B l = a B l , and sends them to Alice. The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017

Anshel-Anshel-Goldfeld Anshel-Anshel-Goldfeld Circumstances: Alice and Bob want to agree on a common key. Platform: let G be a nonabelian group PUBLIC KEYS Alice chooses a 1 , . . . , a l in G and makes them PUBLIC. Bob chooses b 1 , . . . , b k in G and makes them PUBLIC. PRIVATE KEYS Alice chooses A ∈ � a 1 , . . . , a l � . Bob chooses B ∈ � b 1 , . . . , b k � . EXCHANGED INFORMATION Alice computes b ′ 1 , . . . , b ′ 1 = b A k = b A k , and sends them to Bob. Bob computes a ′ 1 , . . . , a ′ 1 = a B l = a B l , and sends them to Alice. The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017

Anshel-Anshel-Goldfeld Anshel-Anshel-Goldfeld Circumstances: Alice and Bob want to agree on a common key. Platform: let G be a nonabelian group PUBLIC KEYS Alice chooses a 1 , . . . , a l in G and makes them PUBLIC. Bob chooses b 1 , . . . , b k in G and makes them PUBLIC. PRIVATE KEYS Alice chooses A ∈ � a 1 , . . . , a l � . Bob chooses B ∈ � b 1 , . . . , b k � . EXCHANGED INFORMATION Alice computes b ′ 1 , . . . , b ′ 1 = b A k = b A k , and sends them to Bob. Bob computes a ′ 1 , . . . , a ′ 1 = a B l = a B l , and sends them to Alice. The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017

Anshel-Anshel-Goldfeld The shared key The shared key is K = [ A , B ] = A − 1 B − 1 AB . Alice determine K via: Write A = w ( a 1 , . . . , a l ) as a word in a 1 , . . . , a l . 1 Compute 2 A − 1 w ( a ′ l ) = A − 1 w ( a B 1 , . . . , a B 1 , . . . , a ′ l ) = A − 1 w ( a 1 , . . . , a l ) B = A − 1 A B = [ A , B ] = K . Bob uses the dual approach to determine K . The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017

Anshel-Anshel-Goldfeld The shared key The shared key is K = [ A , B ] = A − 1 B − 1 AB . Alice determine K via: Write A = w ( a 1 , . . . , a l ) as a word in a 1 , . . . , a l . 1 Compute 2 A − 1 w ( a ′ l ) = A − 1 w ( a B 1 , . . . , a B 1 , . . . , a ′ l ) = A − 1 w ( a 1 , . . . , a l ) B = A − 1 A B = [ A , B ] = K . Bob uses the dual approach to determine K . The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017

Anshel-Anshel-Goldfeld The shared key The shared key is K = [ A , B ] = A − 1 B − 1 AB . Alice determine K via: Write A = w ( a 1 , . . . , a l ) as a word in a 1 , . . . , a l . 1 Compute 2 A − 1 w ( a ′ l ) = A − 1 w ( a B 1 , . . . , a B 1 , . . . , a ′ l ) = A − 1 w ( a 1 , . . . , a l ) B = A − 1 A B = [ A , B ] = K . Bob uses the dual approach to determine K . The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017

Anshel-Anshel-Goldfeld The shared key The shared key is K = [ A , B ] = A − 1 B − 1 AB . Alice determine K via: Write A = w ( a 1 , . . . , a l ) as a word in a 1 , . . . , a l . 1 Compute 2 A − 1 w ( a ′ l ) = A − 1 w ( a B 1 , . . . , a B 1 , . . . , a ′ l ) = A − 1 w ( a 1 , . . . , a l ) B = A − 1 A B = [ A , B ] = K . Bob uses the dual approach to determine K . The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017

Anshel-Anshel-Goldfeld The shared key The shared key is K = [ A , B ] = A − 1 B − 1 AB . Alice determine K via: Write A = w ( a 1 , . . . , a l ) as a word in a 1 , . . . , a l . 1 Compute 2 A − 1 w ( a ′ l ) = A − 1 w ( a B 1 , . . . , a B 1 , . . . , a ′ l ) = A − 1 w ( a 1 , . . . , a l ) B = A − 1 A B = [ A , B ] = K . Bob uses the dual approach to determine K . The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017

Anshel-Anshel-Goldfeld The shared key The shared key is K = [ A , B ] = A − 1 B − 1 AB . Alice determine K via: Write A = w ( a 1 , . . . , a l ) as a word in a 1 , . . . , a l . 1 Compute 2 A − 1 w ( a ′ l ) = A − 1 w ( a B 1 , . . . , a B 1 , . . . , a ′ l ) = A − 1 w ( a 1 , . . . , a l ) B = A − 1 A B = [ A , B ] = K . Bob uses the dual approach to determine K . The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017

Anshel-Anshel-Goldfeld The shared key The shared key is K = [ A , B ] = A − 1 B − 1 AB . Alice determine K via: Write A = w ( a 1 , . . . , a l ) as a word in a 1 , . . . , a l . 1 Compute 2 A − 1 w ( a ′ l ) = A − 1 w ( a B 1 , . . . , a B 1 , . . . , a ′ l ) = A − 1 w ( a 1 , . . . , a l ) B = A − 1 A B = [ A , B ] = K . Bob uses the dual approach to determine K . The Conjugacy Search Problem for Supersoluble Groups Carmine Monetta August 11, 2017