Naccache-Stern knapsack cryptosystem

Note: this is not to be confused with the Naccache-Stern cryptosystem based on the higher residuosity problem.

The Naccache-Stern Knapsack Cryptosystem is an atypical Public Key Cryptosystem developed by David Naccache and Jacques Stern in 1997. This cryptosystem is deterministic, and hence is not semantically secure. This system also lacks provable security.

ystem Overview

This system is based on a type of knapsack problem. Specifically, the underlying problem is this: given integers "c","n","p" and "v"₀,...,"v"_"n", find a vector $x\; in\; \{0,1\}^n$ such that:$c\; equiv\; prod\_\{i=0\}^n\; v\_i^\{x\_i\}\; mod\; p$

The idea here is that when the "v"_"i" are relatively prime and much smaller than the modulus "p" this problem can be solved easily. It is this observation which allows decryption.

Key Generation

To generate a public/private key pair

*Pick a large prime modulus "p".
*Pick a positive integer "n".
*For "i" from 0 to "n", set "p"_"i" to be the "i"th prime, starting with "p"₀ = 2.
*Pick a secret integer "s" < "p"-1, such that gcd("p"-1,"s") = 1.
*Set $v\_i\; =\; sqrt\; [s]\; \{p\_i\}\; mod\; p$. Note: these roots can be calculated using the Pohlig-Hellman algorithm.

The public key is then "p","n" and "v"₀,...,"v"_"n". The private key is "s".

Encryption

To encrypt an "n"-bit long message "m", calculate

:$c\; =\; prod\_\{i=0\}^n\; v\_i^\{m\_i\}\; mod\; p$

where "m"_"i" is the "i"th bit of the message "m".

Decryption

To decrypt a message "c", calculate

:$m\; =\; sum\_\{i=0\}^n\; frac\{2^i\}\{p\_i-1\}\; imes\; left(\; gcd(p\_i,c^s\; mod\; p)\; -1\; ight)$

This works because the fraction

:$frac\{\; gcd(p\_i,c^s\; mod\; p)\; -\; 1\; \}\{p\_i\; -\; 1\}$

is 0 or 1 depending on whether "p"_"i" divides "c"^"s" mod "p".

References

[http://citeseer.ist.psu.edu/205954.html Original Paper]

[http://eprint.iacr.org/2008/119.pdf Recent bandwidth improvement]