Algebraic manipulation detection codes from algebraic curves

An application specific integrated circuit tangibly encoding a program of instructions executable by the integrated circuit to perform a method for protecting encrypted data from algebraic manipulation, the method comprising the steps of:

• receiving a data word sεKd, at the application specific integrated circuit, from a communications channel to be protected, wherein K is a finite field of q elements for a square prime power q and a predetermined d;
• the application specific integrated circuit randomly selecting two integers a ε{0, . . . , q−1} and b ε{0, . . . , √q−1};
• the application specific integrated circuit executing instructions for finding a point (α, β) on a Hermitian curve over a field Fq that corresponds to the randomly selected integers (a, b) from a mapping (a, b)(α, β)=ua, ua√q+1z+vb), wherein
• u
• a
• :=
• {
• 0
• if
• ⁢
• ⁢
• a
• =
• 0
• ,
• γ
• 1
• a
• -
• 1
• otherwise
• ,
• ⁢
• V
• b
• :=
• {
• 0
• if
• ⁢
• ⁢
• b
• =
• 0
• ,
• γ
• 2
• b
• -
• 1
• otherwise
• ,
• and z is a pre-selected element of the field Fq of unit trace, and wherein γ1 is a pre-selected fixed primitive element of the field Fq and γ2 is a pre-determined fixed element of the field Fq which is a primitive element of a field F√q ⊂ Fq;
• and
• calculating, by the application specific integrated circuit, a sum
• fs(α,β)=αid+1βjd+1+Σk=1dskαikβjk
• for an appropriate set of d+1 integers pairs I={(ik,jk)}k=1d+1,
• outputting, by the application specific integrated circuit, a triple (s, (α, β),fs(α, β)) as an encoded word, wherein said encoded word is protected from side-channel attacks during transmission.

An application specific integrated circuit tangibly encoding a program of instructions executable by the integrated circuit to perform a method for protecting encrypted data from algebraic manipulations, the method comprising the steps of:

• receiving a triple (s, (α, β), γ), at the application specific integrated circuit, from a communications channel that is an element of Fqd×Fq2×Fq, wherein s is an encoded data word, and q and d are predetermined constants;
• the application specific integrated circuit decoding the encoded data word using algebraic manipulation detection (AMD) Hermitian codes by determining whether β√q+β=α√q+1;
• and
• if β√q+β=α√q+1, further determining, by the application specific integrated circuit, whether a sum fs(α, β)=γ, wherein
• fs(α,β)=αid+1βjd+1+Σk=1dskαikβjk
• for an appropriate set of d+1 integers pairs I={(ik,jk)}k=1d+1,
• wherein if either β√q+β≠α√q+1 or fs(α,β)≠γ, declaring an error, otherwise declaring that no error was injected into s.