How to Garble Arithmetic Circuits

Yao's garbled circuit construction transforms a boolean circuit C : {0, 1}
ⁿ
→ {0, 1}
^m
into a "garbled circuit" Ĉ along with n pairs of k-bit keys, one for each input bit, such that Ĉ together with the n keys corresponding to an input x reveal C(x) and no additional information about x. The garbled circuit construction is a central tool for constant-round secure computation and has several other applications. Motivated by these applications, we suggest an efficient arithmetic variant of Yao's original construction. Our construction transforms an arithmetic circuit C : ℤ
ⁿ
→ ℤ
^m
over integers from a bounded (but possibly exponential) range into a garbled circuit Ĉ along with n affine functions L
_i
: ℤ → ℤ
^k
such that Ĉ together with the n integer vectors L
_i
(x
_i
) reveal C(x) and no additional information about x. The security of our construction relies on the intractability of the learning with errors (LWE) problem.