a) In our lecture we discussed how Alice can use a binary Merkle tree to commit to a set of messages S = {m₁, M₂, M3,..., mg) by providing Bob with one commitment value which is the Merkle root. Later she can prove to Bob that some m, is in S by sending him an authentication path containing at most [log₂ n] values. Now consider that Alice decided to do the same but using ternary Merkle trees where every non-leaf node has three children (instead of 2). The hash value for every non-leaf node is computed as the hash of the concatenation of the values of its children. i) ii) Suppose S= {m₁, M₂, M3,..., mg). Explain how Alice computes a one commitment value to all the messages in S using the above described ternary Merkle tree. You may use pictorial illustration. Explain how Alice later proves to Bob that m4 is in the set S which is committed to in the previous step. iii) iv) Consider generalizing the problem of accumulator commitments using ternary Merkle trees where S contains n messages. What is the size of the authentication path that proves that some mi is in S as a function of n? For a very large n, is the authentication path using a ternary Merkle tree shorter or longer than when using a binary Merkle tree? Please explain.
a) In our lecture we discussed how Alice can use a binary Merkle tree to commit to a set of messages S = {m₁, M₂, M3,..., mg) by providing Bob with one commitment value which is the Merkle root. Later she can prove to Bob that some m, is in S by sending him an authentication path containing at most [log₂ n] values. Now consider that Alice decided to do the same but using ternary Merkle trees where every non-leaf node has three children (instead of 2). The hash value for every non-leaf node is computed as the hash of the concatenation of the values of its children. i) ii) Suppose S= {m₁, M₂, M3,..., mg). Explain how Alice computes a one commitment value to all the messages in S using the above described ternary Merkle tree. You may use pictorial illustration. Explain how Alice later proves to Bob that m4 is in the set S which is committed to in the previous step. iii) iv) Consider generalizing the problem of accumulator commitments using ternary Merkle trees where S contains n messages. What is the size of the authentication path that proves that some mi is in S as a function of n? For a very large n, is the authentication path using a ternary Merkle tree shorter or longer than when using a binary Merkle tree? Please explain.
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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![a) In our lecture we discussed how Alice can use a binary Merkle tree to commit to a set of
messages S = {m₁, M₂, M3,..., mg} by providing Bob with one commitment value which is
the Merkle root. Later she can prove to Bob that some m, is in S by sending him an
authentication path containing at most [log₂ n] values. Now consider that Alice decided to
do the same but using ternary Merkle trees where every non-leaf node has three children
(instead of 2). The hash value for every non-leaf node is computed as the hash of the
concatenation of the values of its children.
i)
ii)
Suppose S = {m₁, M₂, M3, ..., mg}. Explain how Alice computes a one commitment
value to all the messages in S using the above described ternary Merkle tree. You
may use pictorial illustration.
Explain how Alice later proves to Bob that m4 is in the set S which is committed to
in the previous step.
iii)
iv)
Consider generalizing the problem of accumulator commitments using ternary
Merkle trees where S contains n messages. What is the size of the authentication
path that proves that some mi is in S as a function of n?
For a very large n, is the authentication path using a ternary Merkle tree shorter or
longer than when using a binary Merkle tree? Please explain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc73c7743-bc0a-4309-8531-6ea8e780a6d6%2F4fd867ec-0469-4def-b057-f469faf89b5d%2Ft61m7tc_processed.png&w=3840&q=75)
Transcribed Image Text:a) In our lecture we discussed how Alice can use a binary Merkle tree to commit to a set of
messages S = {m₁, M₂, M3,..., mg} by providing Bob with one commitment value which is
the Merkle root. Later she can prove to Bob that some m, is in S by sending him an
authentication path containing at most [log₂ n] values. Now consider that Alice decided to
do the same but using ternary Merkle trees where every non-leaf node has three children
(instead of 2). The hash value for every non-leaf node is computed as the hash of the
concatenation of the values of its children.
i)
ii)
Suppose S = {m₁, M₂, M3, ..., mg}. Explain how Alice computes a one commitment
value to all the messages in S using the above described ternary Merkle tree. You
may use pictorial illustration.
Explain how Alice later proves to Bob that m4 is in the set S which is committed to
in the previous step.
iii)
iv)
Consider generalizing the problem of accumulator commitments using ternary
Merkle trees where S contains n messages. What is the size of the authentication
path that proves that some mi is in S as a function of n?
For a very large n, is the authentication path using a ternary Merkle tree shorter or
longer than when using a binary Merkle tree? Please explain.
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