Battery History

The following are notes on battery history

• History of Batteries
• 2000 years ago Baghdad battery
• created in Mesopotamia 
• Jar of ceramics created with a speculated structure of copper with walls of iron
• if filled with unknown composition of electrolytes meets the requirements for a battery
• Maybe used to plate gold onto silver?
• 200 Years Ago Animal Electricity
• Galvani while doing a dissection a steel scalpel touched a brass hook that was holding a frog leg in place
• Battery composition
• Steel scalpel has iron 
• brass hook has copper
• preservatives acted as electrolytes
• First Battery
• Voltaic Pile 
• set of individual galvanic cells placed in series
• copper and zinc stacked together with electrolytes sandwiching the two metals
• History of electrochemical energy storage
• Galvani
• Volta - Voltaic Pile 
• Daniell - Zn/Cu Cell
• Faraday - Basic Principles of electrochemistry
• Grove - H2/02 cell
• Plante - Lead Acid Cell
• Leclanche - Zn Carbon dry cell
• Edison Zn/Ni and Fe/Ni cells
• Modern Batteries
• modern development started around half a century ago
• Solid Electrolytes
• 1966 Yao and Kummer Beta Alumina Solid Electrolyte: Fast Na Ion Conductor
• birthed solid state ionics
• Rubidium silver iodide
• Internal Phenomena
• Whittingham steele insertion reaction electrodes
• created Le/TiS2 cells
• Sony commercialized the first Lithium ion batteries

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Notes - The ElGamal Public Key Cryptosystem

The following are notes from Introduction to Cyrptography with Coding Theory.

• ElGamal Public Key Cryptosystem
• Alice wants to send a message m to Bob
• Bob chooses a large prime p and a primitive root α
• Bob chooses a secret integer a and computes B ≡ α ^a (mod p)
• Make public (p, α, B) and is Bob's public key
• Alice
• downloads (p, α, B)
• Chooses a secret random integer k and computes r ≡ α ^k (mod p)
• Computes t ≡ B ^k m (mod p)
• sends the pair (r,t) to B
• Bob decrypts by computing tr ^-a ≡ m (mod p)
• Security
• depends on a being kept secret which is reliant on the idea that discrete logs are difficult to compute

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Notes - Discrete Logarithms

The following are notes from Introduction to Cyrptography with Coding Theory.

• RSA Algorithm relies on the difficulty of factoring, but we can also use the difficulty of discrete logs
• Fix a prime p. Let α and Β be nonzero integers mod p and suppose
• Β ≡ α^x (mod p)
• It is difficult to find x assuming we choose a large enough prime
• α is taken to be a primitive root mod p so that every Β is a power of α (mod p)
• The size of the largest primes for which discrete logs can be computed is usually about the same size as the largest integers that could be factored
• This means that discrete logs are another example of one-way functions
• where f(x) is easy to compute but given y it is computationally infeasible to find x

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Notes - An Application to Treaty Verification

The following are notes from Introduction to Cyrptography with Coding Theory.

• This method will essentially describe the RSA signature Scheme
• Example
• Countries A and B have signed a nuclear test ban treaty
• Country A wants to verify the treaty by placing sensors in B
• Country A wants to ensure sensors are sending correct data without tampering
• Country B wants to look at the message before its being sent to ensure that espionage data is not being transmitted
• A chooses n = pq to be the product of two large primes and chooses encryption/decryption exponents e and d
•  n and e are given to B but p q and d are kept secret
• sensor collects data x and uses d to encrypt x to y ≡ x^d (mod n)
• both x and y are sent to country B to check that y^e ≡ x (mod n) and then to Country A to ensure that the number x has not been modified. If x was modified it would be the same difficulty as decrypting the rsa message x

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Notes - The RSA Algorithm

The following are notes from Introduction to Cryptography with Coding Theory.

• Public Key Cryptosystem
• Presented by Diffie-Hellman
• Goal - allow Alice to send a message to Bob without previous contact and without the use of a courier to exchange a key but protect the message from Eve a potential attacker
• The RSA Algorithm
• Bob chooses secret primes p and q and computes n = pq
• Bob chooses e with gcd(e, (p-1)(q-1)) = 1
• Bob computes d with de ≡ 1 (mod(p-1)(q-1))
• Bob makes n and e public, and keeps p, q, d secret
• Alice encrypts m as c ≡ m^e (mod n) and sends c to Bob
• Bob decrypts by computing m ≡ c^d (mod n)
• Example of the RSA Algorithm
• p = 885320963, q = 238855417
• n = p * q = 211463707796206571
• Let the encryption exponent be e = 9007
• Let the sample message m be 30120
• c  ≡ m^e ≡ 30120⁹⁰⁰⁷ ≡ 113535859035722866 (mod n)
• Where c stands for the ciphertext Alice is sending to Bob
• Bob knows p and q so he knows (p-1)(q-1) then he uses the extended euclidean algorithm to compute d such that de ≡ 1  mod (p-1)(q-1)
• d = 116402471153538991
• Bob computes c^d ≡ 113535859035722866^{116402471153538991} ≡ 30120 (mod n) to obtain the original message

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