A method for converting an element of a finite field of characteristic q, wherein q is equal to 2, from a representation in a first basis defined by a first irreducible polynomial to a representation in a second basis defined by a second irreducible polynomial, the method being used to support secure communication between a device of a first party and a device of a second party, the device of the first party implementing a cryptographic system using the representation in said first basis and the device of the second party implementing a cryptographic system using the representation in said second basis, said method comprising the steps of: obtaining said element of said finite field represented in said first basis as a polynomial a ( x) from said cryptographic system implemented on the device of the first party; specifying a root r of said second irreducible polynomial corresponding to the required conversion; evaluating said polynomial a ( x) at said root r to obtain a representation a ( r) of a ( x) in said second basis for use in said cryptographic system; said evaluation being characterised by the steps of: partitioning said polynomial a ( x) into two component polynomials, b ( x) and c ( x) such that said polynomial a ( x) is recoverable by combining said two component polynomials using operations of multiplication by x and exponentiation by q; obtaining values of each of said component polynomials by evaluating each of said component polynomials at said root r; computing the value of a ( r) from said values of each of said component polynomials at said root r, using operations of multiplication by rand exponentiation by q wherein said component polynomials are combined using the formula a ( r) = ( b ( r)) 2 + r ( c ( r)) 2; and providing said representation a ( r) in said second basis to said cryptographic system implemented on the device of the second party. obtaining said element of said finite field represented in said first basis as a polynomial a ( x) from said cryptographic system implemented on the device of the first party; specifying a root r of said second irreducible polynomial corresponding to the required conversion; evaluating said polynomial a ( x) at said root r to obtain a representation a ( r) of a ( x) in said second basis for use in said cryptographic system; said evaluation being characterised by the steps of: partitioning said polynomial a ( x) into two component polynomials, b ( x) and c ( x) such that said polynomial a ( x) is recoverable by combining said two component polynomials using operations of multiplication by x and exponentiation by q; obtaining values of each of said component polynomials by evaluating each of said component polynomials at said root r; computing the value of a ( r) from said values of each of said component polynomials at said root r, using operations of multiplication by rand exponentiation by q wherein said component polynomials are combined using the formula a ( r) = ( b ( r)) 2 + r ( c ( r)) 2; and providing said representation a ( r) in said second basis to said cryptographic system implemented on the device of the second party. partitioning said polynomial a ( x) into two component polynomials, b ( x) and c ( x) such that said polynomial a ( x) is recoverable by combining said two component polynomials using operations of multiplication by x and exponentiation by q; obtaining values of each of said component polynomials by evaluating each of said component polynomials at said root r; computing the value of a ( r) from said values of each of said component polynomials at said root r, using operations of multiplication by rand exponentiation by q wherein said component polynomials are combined using the formula a ( r) = ( b ( r)) 2 + r ( c ( r)) 2; and providing said representation a ( r) in said second basis to said cryptographic system implemented on the device of the second party.