Tag Archives: Blockchain

51% Blockchain attack

One of the predicted, and proved feasible, attack to the blockchain technology is the 51% attack. There are several forms of this attack including weakness in the blockchain algorithm itself that allowed easier than usual forking of chain to dishonestly win the popular vote mechanism. In other cases, the perpetrator has to amass more hashing power than all other nodes in the blockchain combined to corrupt the distributed ledger, and hence the name of “51%”.

An interesting mathematical study on the effectiveness of this attack is based on the regularized incomplete beta function to calculate the probability of success of a perpetrator.

incompletebeta

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Using TI Nspire to explore mathematics behind blockchain technology

The TI Nspire is a great tool for exploring mathematics through its calculation and graphing capability. One of the emerging technologies that is based on mathematics is blockchain. It gained popularity through bitcoin that caused much debate and controversy in the field of banking, economics and finance. Until recently more and more established research and technology firms started to look at it seriously and its underlying core technology, blockchain, is gaining momentum for being adopted by traditional financial institutions.

In a previous installment, the property of the Elliptic Curve is explored using TI Nspire. As shown in the dynamic graph below, the curve exhibit several properties that form the two basic operations of asymmetric encryption – point addition and point doubling – for public and private key pair generation.

ecanim

Using a=0, and b=7 as in Bitcoin, the two properties are basically illustrated in the following graphs.

Point Addition
blockchainecc1c

Point Doubling
blockchainecc2c2

However, in reality, the Elliptic Curve Digital Signature Algorithm (ECDSA) algorithm to generate public and private key relied also on another mathematical concept known as the finite field. This is basically a limit imposed on the numbers that are available for use in the calculation, and in this case, positive integers from a modulo calculation. The prime modulo for Bitcoin (as in secp256k1) is set to  2256 – 232 – 29 – 28 – 27 – 26 – 24 – 1. Having this in place, the graph will not look like the above but some scattered points on a fixed region, and overflows will wrap around. However, the symmetry will still be preserved and recognized visually on graph.

With this mathematical backed technology as the foundation, blockchain can provides open ledger for secure transaction service.