System and method for efficient basis conversion

This invention describes a method for evaluating a polynomial in an extension field FqM, wherein the method comprises the steps of partitioning the polynomial into a plurality of parts, each part is comprised of smaller polynomials using a q-th power operation in a field of characteristic q; and computing for each part components of q-th powers from components of smaller powers. A further embodiment of the invention provides for a method of converting a field element represented in terms of a first basis to its representation in a second basis, comprising the steps of partitioning a polynomial, being a polynomial in the second basis, into a plurality of parts, wherein each part is comprised of smaller polynomials using a q-th power operation in a field of characteristic q; evaluating the polynomial at a root thereof by computing for each part components of q-th powers from components of smaller powers; and evaluating the field element at the root of the polynomial.

TECHNICAL FIELD

This invention relates to the field of cryptographic systems, and conversion of elements between bases used in such systems.

BACKGROUND OF THE INVENTION

It is well known that there is only one finite field of any given order, but that there are many different representations. When an extension field is built by adjoining a root of an irreducible polynomial to the ground field, the choice of irreducible affects the representation of the extension field. In general if Fqmis the finite field, where q is a prime and Fqis the ground field over which it is defined, the elements of the finite field can be represented in a number of ways depending on the choice of basis. In order to interoperate, cryptographic systems employing finite fields often need to establish a common representation. In addition to the choice of irreducible polynomial, finite fields can also be represented by either polynomial or normal basis. A polynomial basis represents elements of Fqmas linear combinations of the powers of a generator element x: {x0, x1, . . . , xm−1}. A normal basis representation represents elements as linear combination of successive q-th powers of the generator element x: {xq0, xq1, . . . , xqm−1}. Each basis has its own advantages, and cryptographic implementations may prefer one or the other, or indeed specific types of irreducible polynomials, such as trinomials or pentanomials.

To support secure communication between devices using different representations, basis conversion, which changes the representation used by one party into that used by another party is generally required.

Basis conversion often entails the evaluation of a polynomial expression at a given finite field element. If an element a, when represented as a polynomial, is given as a(x)=Σaiximod f(x), where f(x) is an irreducible, in one basis, then the conversion of the element a into a new representation using another irreducible polynomial requires that a be evaluated at r, where r is a root of the new irreducible polynomial in the field represented by f(x), then a(r) is the element a in the new representation. Any of the conjugates of r (the other roots of the new irreducible) will also induce equivalent, but different representations.

There is a need for an efficient method for evaluating these polynomials, for application to basis conversion.

SUMMARY OF THE INVENTION

In accordance with this invention there is provided a method for evaluating polynomials in an extension field comprising the steps of: partitioning the polynomials into a plurality of parts, such that each part may be computed from smaller polynomials using a q-th power operation in a field of characteristic q.

In accordance with a further embodiment of the invention there is provided a method for evaluating a polynomial in an extension field comprising the steps of computing components of the q-th powers from components of smaller powers.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

In a first embodiment of the invention, we exemplify a specific case F2mof the general field Fqm, then in order to evaluate a field element a(x)=Σaixiin F2m, it is noted that approximately one half of the exponents xiare even powers. These powers can be efficiently calculated from lower degree exponents of odd powers. For example, the powers for i=2,4,6,8,10 can be calculated by squaring the powers for i=1,2,3,4,5, respectively. This approach does not apply to the odd powers, but if we partition a into even and odd powers, e.g. a(x)=aeven(x)+aodd(x), and if we factor x from the odd powers, then a will be represented by a sum of even powers and x times a sum of even powers.

a⁡(x)=⁢(a0+a2⁢x2+a4⁢x4+…⁢)+x(a1+a3⁢x2+a5⁢x4+…⁢)=⁢aeven⁡(x)+xaeven′⁡(x)
where a′even(x) is the even-powered polynomial derived by factoring x from aodd(x).

In a field of characteristic 2, F2msquaring is a linear operation, which allows aevenand a′evento be expressed as squares of polynomials of smaller (roughly one half) degree. Explicitly, defining b(x)=a0+a2x+a4x2+a6x3+ . . . and c(x)=a1+a3x+a5x2+a7x3+ . . . , then a(x) can be expressed as a(x)=(b(x))2+x(c(x))2. Now b and c have approximately half the degree of the original polynomial a to be evaluated.

Evaluation of b (and c) can (recursively) be made by further dividing the polynomial into odd and even parts. The odd part can be shifted to an even power by factoring out x and expressing the result as a combination of squares of lower degree polynomials. At each application of the partitioning and shifting two squaring operations and one multiplication by x are required.

At some point, the recursive partitioning can be halted and the component polynomials evaluated via one or more methods.

Note that although we have described the method for the extension fields over F2, similar application can be made to other characteristics if the polynomials are divided into more components. For example, for the extension held over F3, the polynomial a(x) might be divided into a(x)=a0 mod 3+a1 mod 3+a2 mod 3, where
a0 mod=(a0+a3x3+a6x6. . . )a1 mod 3=(a1x+a4x4+a7x7. . . ), and a2 mod 3=(a2x+a5x5+a8x8. . . ).
In general, for extension fields over Fqthe polynomial may be divided into q parts. Even over characteristic 2, the polynomial a might be broken up into more than 2 polynomials, at the cost of more multiplications by x or powers of x.

As an example of a preferred embodiment at a given size, consider conversion of polynomial representations over F2163. An element of this field is represented by the polynomial
a(x)=a0+a1x+a2x2+ . . . a162x162.
The first branching divides a(x) into:

An exemplary method will now be described for efficiently evaluating a component polynomial for extension fields over F2.

At the leaf, a component polynomial a(x)=Σaiximust be evaluated at a root of a new irreducible. Again, approximately one half of these terms will be even and can be calculated efficiently from odd powers. These odd powers will either be stored, for explicit basis conversion, or calculated explicitly by multiplication. If, for example a(x) is guaranteed to be of degree not greater than 10 (which may be the case if certain evaluation tree depths are employed), then a(x) can be evaluated from the powers 1,3,5,7,9, which are either stored or calculated. Squarings can be reduced by grouping coefficients together. This is shown schematically inFIG. 2(a) where a notional table is constructed to show the relationship between the stored or evaluated odd powers of r and the higher degree even powers of r. Thus, consider the first row in which r2, r4and r8are derived by squaring r1, similarly, r6is derived by squaring r3and r10is derived by squaring r5. It is to be noted that in this example, powers of 2 are used.

Turning back toFIG. 2(a), however, the notional table may be used as shown schematically inFIG. 2(b). Thus, assume an accumulator is set initially to 0. Since we are using an extension field over F2the coefficients aiare either 0 or 1. First, if a8is 1, then r1is added to the accumulator, which consists of a copying operation in a processor. Next, the accumulator is squared. Next, if a4is 1, then r1is added into the accumulator. Again, the accumulator is squared. Now, if a2, a6, a10are one (1) then r1, r3, r5are added into the accumulator respectively. Again, the accumulator is squared. Finally, if a0, a1, a3, a5, a7, a9are set (1), then r0, r1, r3, r5, r7, r9are added into the accumulator. This completes the evaluation o a(x) at r, requiring three squares and the initial evaluation of r0,r1,r3,r5,r7,r9, which can be reused at another leaf evaluation.

It will be apparent to those skilled in the art that the precomputed values technique can equally well be applied to multiplication techniques.

For polynomials of larger degrees, similar evaluations can be made from evaluation of odd powers. First, the coefficients of those exponents with the largest powers of 2 are added into the accumulator according to whether the polynomial to be evaluated has non-zero coefficients at those powers, then the result is squared. Next, powers divisible by one less power of 2 are added in as required by polynomial evaluation. Accumulation and squaring continues until the odd powers themselves are added in as required by the polynomial under evaluation.

InFIGS. 3(a) and3(b), a similar evaluation is exemplified for an extension field over F3and for a polynomial of degree no greater than 17. Note that in this embodiment, the coefficients aimay take a value 0, 1, or 2. Thus, the powers are added with the required coefficients. In general then, for an extension field over Fq, powers of q are used to construct the notional table and evaluation of the polynomial proceeds by accumulation and q powering until all required powers in the polynomial are added in as required by the polynomial being evaluated.

An application of the above method to basis conversion may be illustrated as below. Given a field F231and a pair of bases having respective irreducible f1and f2and if f1=x31+x6+1; and f2=x31+x3+1. Then, a root of f1in the field represented by f2is given by r=x26+x24+x23+x22+x19+x17+x12+x11+x9+x8+x6+x5+x3+x2. Now, to convert an element a(x)=af1in the first basis to a representation in the second basis af2(that is to basis defined by f2) we proceed as follows. Let

a⁡(x)=∑i=030⁢⁢ai⁢xi
in general. For this example, we choose a specific element:
a(x)=x30+x29+x28+x27+x25+x22+x20+x19+x14+x13+x12+x11+x10+x8+x7+x6+x3+x0

We assume a three level evaluation tree which is constructed in accordance with the present invention as shown inFIG. 4. At the bottom level of the tree (the leaf nodes), we require the following powers of r: r0, r1, r2. . . r7. The odd powers are calculated r1, r3, r5, and r7(by squaring r and 3 multiplications by r2).

When a above is decomposed in the tree, the leaf nodes are:
L0=(r7+r5+r3+r2+1)2
L1=r(r7+r5+r3+r2+r)2
L2=(r7+r3)2
L3=r(r6+r5+r4+r2+r+1)2
To evaluate the leaf node L0, we will evaluate the component polynomial, then square it and, when appropriate, further multiply its value by r to obtain the value of the leaf node:0) zero A1) add r1to A, square A, now A=r22) add in r0, r3, r5, r7to A3) square A=L0
For L1, we will0) zero A1) add r1to A2) square A3) add r1, r3, r5, r7, to A4) square A5) multiply A by r=L1
for L20) zero A1) add in r3, r72) square A=L2
for L30) zero A1) add in r12) square A=r23) add in r1, r34) square A=r6+r4+r25) add in r0, r1, r5A=r6+r5+r4+r2+r+16) square A7) multiply A by r=L3
Now a(r) is built by evaluating the tree M0=(L0+L1)2, M1=r(L2+L3)2. Finally, a(r)=T0=M0+M1.

Thus, it may be seen that his method may be applied to various cryptographic schemes such as key exchange schemes, signature schemes and encryption schemes.

Although the invention has been described with reference to certain specific embodiments, various modifications thereof will be apparent to those skilled in the art without departing from the spirit and scope of the invention as outlined in the claims appended hereto. For example, the invention may be applied to basis conversion wherein the bases to be converted between are an optimal normal basis and a polynomial basis.