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Date: Mon, 8 Sep 2014 12:00:10 +0800
From: Ben Harris <>
Subject: Generalised password hashing for arbitrary proof of work

I've been looking at Cuckoo Cycle ( as a proof
of work function recently. The tl;dr is that it is a graph theoretic PoW
that is dominated by memory access time. In the interests of discussion,
I'll suggest a replacement to Catena's lambda-BRH for a Cuckoo Cycle proof
of work function.

Cuckoo cycle has variables N, M and H, takes an arbitrary input and outputs
H 64-bit positive integers (value less than M). N is the number of nodes in
the graph, M the number of edges, and H the target length of a hash
collision cycle.

In the standard case, M is N/2 and H is 42. This way only allows for a 2.2%
chance of solution, for password hashing >99% chance of solution is
desired. At M = 0.75*N, the chance is >99% and the memory load is 4 reads
and 1.2 writes per M. Memory requirements are M bits + some fraction of N.
As the chance of all 42 edges in the cycle falling only within one half of
the nodes is extremely small (0.5^42), attempting the process with even M/2
memory is practically impossible. The author estimates 67% of the runtime
is DRAM memory latency.

lambda-BRH as a function takes a garlic value and an hash-size input, then
outputs a hash-size output.
The proposed CC_0.75_42 takes the same inputs and outputs. The value N is
hash-size*2^garlic, M is floor(0.75*N) and H is 42. To convert the 42
64-bit integers into a hash-size output, sort them ascending, concatenate
together, prepend with the input and hash. If more than one solution is
found, use the one with the smallest first value (when sorted ascending).
If no solution is found (extremely unlikely) restart with a different salt
value (only needs to be done the first time the salt is created).

This keeps all the extra features of Catena, but loses the secret
independent memory access.

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